PARAMETER ESTIMATION FOR ORNSTEIN-UHLENBECK ...

Communications on Stochastic Analysis Vol. 1, No. 2 (2007) 175-192

PARAMETER ESTIMATION FOR ORNSTEIN-UHLENBECK ´ PROCESSES DRIVEN BY α-STABLE LEVY MOTIONS YAOZHONG HU AND HONGWEI LONG Abstract. The parameter estimation theory for stochastic differential equations driven by Brownian motions or general Lévy processes with finite second moments has been well developed. In this paper, we consider the parameter estimation problem for Ornstein-Uhlenbeck processes driven by α-stable Lévy motions. The classical maximum likelihood method does not apply in this context because the likelihood ratio does not exist. We shall use the trajectory fitting method combined with the weighted least squares technique. We discuss the consistency and the asymptotic distributions of our estimators for general weights in both the ergodic and the non-ergodic cases.

1. Introduction and Notation We consider the Ornstein-Uhlenbeck processes X = {Xt , t ≥ 0} determined by the following linear stochastic differential equation ( dXt = −θXt dt + mdt + σdZt , t ≥ 0, (1.1) X0 = x, where θ, m and σ are given constants and Zt is a given standard α-stable Lévy motion. The basic probability space is (Ω, F, P) equipped with a right continuous and increasing family {Ft , t ≥ 0} of σ-algebras. The expectation on this probability space is denoted by E. For some technical reason we also assume that 1 < α < 2. Suppose we don’t know the parameter θ (m and σ known). We have observation of the process X = {Xt , 0 ≤ t ≤ T } up to the time instant T . We are interested in estimating the unknown parameter θ. When Zt is replaced by a standard Brownian motion, the parameter estimation for θ has been extensively studied by using classical maximum likelihood method or by using the least squares technique (see Liptser and Shiryaev [8]). However, the naive classical maximum likelihood estimator (MLE) is no longer valid in our setting because the explicit density functions are not available and the Girsanov measure transformation is not well defined for the α-stable Lévy motions. In the next section we shall sketch how to use the least squares technique to study the estimator of θ when Zt is a Brownian 2000 Mathematics Subject Classification. Primary 62M05, 62F12; secondary 60F15, 60H10, 60G52. Key words and phrases. Lévy processes, generalized Ornstein-Uhlenbeck processes, weighted trajectory fitting estimator, invariant measures, consistency, asymptotic distribution of the weighted TFE.. Hu is supported by the National Science Foundation under Grant No. DMS0504783. Long is supported by FAU Start-up funding at the C. E. Schmidt College of Science. 175

176

YAOZHONG HU AND HONGWEI LONG

motion and explain why the naive least squares technique can no longer apply in our case. To find a consistent estimator θˆ of θ for (1.1) we shall use the trajectory fitting method combined with the weighted least squares technique. The trajectory fitting method was first proposed by Kutoyants [6] as a numerically attractive alternative to the well-developed maximum-likelihood estimators for continuous diffusion processes (see Dietz and Kutoyants [2], [3], Dietz [1], and Kutoyants [7]). To obtain our estimator we introduce Z t At = Xs ds, t > 0. 0

The equation (1.1) can be written as Xt = x − θAt + mt + σZt . Let wt be a deterministic positive (weight) function. Multiply the above equation by wt we have wt Xt = wt x − θwt At + mtwt + σwt Zt . The weighted trajectory fitting estimate (TFE) of θ is to minimize Z T 2 |wt Xt − (wt x − θwt At + mtwt )| dt. 0

It is easy to see that the minimum is attained when θ is given by RT 2 w (Xt − x − mt)At dt ˆ . θT = − 0 t R T wt2 A2t dt 0

(1.2)

First we shall prove the strong consistency of θˆT : lim θˆT = θ

T →∞

P − a.s.

[In the rest of the paper, when we don’t specify the convergence we always mean the almost sure convergence.] Once we have established the above result we can study the asymptotic distributions of θˆT . Namely, we will find a functional κ(w, T ) such that κ(w, T ) θˆT − θ converges in distribution to a α-stable random variable Ξ (independent of T and w). This means that θˆT − θ ≈

1 Ξ as T → ∞. κ(w, T )

1 −δ If κ(w,T for some positive δ, namely, ) is of the order of T functional F (w) of w, then we have

Tδ κ(w,T )

converges to a

1 θˆT − θ ≈ F (w) δ Ξ as T → ∞. T F (w) is called the leading coefficient. We prefer to have a smaller value of F (w) for a given class of weight functions.

PARAMETER ESTIMATION FOR O-U PROCESSES

177

In the following we demonstrate how to view the stable process from general context of Lévy process. Namely, we consider ( dXt = −θXt dt + dLt , t ≥ 0, (1.3) X0 = x, where θ is an unknown parameter, {Lt , t ≥ 0} is a one-dimensional Lévy process. Lévy processes are closely related to stable distributions. A random variable Z is said to have a stable distribution with index of stability α ∈ (0, 2], scale parameter σ ∈ (0, ∞), skewness parameter β ∈ [−1, 1], and location parameter µ ∈ (−∞, ∞) if it has characteristic function (c.f.) of the following form: φZ (u)

= E exp{iuZ} exp −σ α |u|α 1 − iβsgn(u) tan απ , if α 6= 1, 2 + iµu = exp −σ|u| 1 + iβ π2 sgn(u) log |u| + iµu , if α = 1.

For the random variable Z distributed according to the rule described above we use the notation Z ∼ Sα (σ, β, µ). When µ = 0, we say Z is strictly stable. If in addition β = 0, we call Z symmetric α-stable. We refer to Samorodnitsky and Taqqu [10], Janicki and Weron [4] and Sato [11] for more details on stable distributions. Suppose that {Lt , t ≥ 0} is a Lévy process generated by the triplet (0, ρ, λ), i.e. the distribution of Lt has characteristic function ( ) Z iuLt iux φLt (u) = E[e ] = exp itλu + t (e − 1 − iux1D (x))ρ(dx) , u ∈ R , R \{0}

(1.4)

where D = {x : |x| ≤ 1} and ρ is the Lévy measure given by c2 c1 ρ(dx) = 1+α 1(0,∞) (x)dx + 1+α 1(−∞,0) (x)dx, x |x|

(1.5)

where 1 < α < 2, c1 ≥ 0, c2 ≥ 0, and c1 + c2 > 0. It is easy to see that (1.4) can be written as ( ! ) Z h πα i α α xρ(dx) u − tσ |u| 1 − βsgn(u) tan φLt (u) = exp it λ + , 2 |x|>1

where σ α = −(c1 + c2 )Γ(−α) cos(πα/2) and β = (c1 − c2 )/(c1 + c2 ). Then, by the Itˆ o-Lévy decomposition, we have Z tZ Z tZ e (ds, dx) + Lt = λt + xN xN (ds, dx), (1.6) 0

|x|1

m=λ+ Denote

Z tZ

xρ(dx) |x|≥1

c1 − c 2 . α−1

R \{0}

e (ds, dx). xN

Then Z˜t is a α-stable Lévy motion and Z˜t − Z˜s ∼ Sα (σ(t − s)1/α , β, 0) for any 0 ≤ s < t < ∞. We can renormalize Z˜t and define Zt = Z˜t /σ. Then we can easily see that {Zt , t ≥ 0} is a standard α-stable Lévy motion (see Janicki and Weron [4]) so that Z1 has a stable distribution Sα (1, β, 0). It is clear that Lt = mt + σZt and E[Lt ] = mt. The paper is organized as follows. In Section 2, we sketch the least squares method in classical setting. In Section 3, we prove the strong consistency and discuss the asymptotic distribution of the weighted TFE in the ergodic case. In Section 4, we shall establish some results for the weighted TFE in the non-ergodic case. 2. Classical Least Squares Technique Consider the Langevin equation dXt = −θXt dt + dBt ,

(2.1)

where (Bt , t ≥ 0) is a standard Brownian motion and θ is the unknown parameter to be estimated from the observation (Xt , 0 ≤ t ≤ T ). To explain the least squares technique we write formally X˙ t = −θXt + B˙ t and we minimize Z T Z 2 ˙ |Xt + θXt | dt = 0

The minimizer is given by

T 0

X˙ t2 dt + 2θ

Z

T

Xt X˙ t dt + θ2

0

RT Xt dXt ˆ θT = − R0 T Xt2 dt 0

Z

T 0

Xt2 dt .

(2.2)


179

RT (The meaningless term 0 X˙ t2 dt does not appear). To show the least squares estimator θˆT converges to θ, we have RT RT RT RT Xt dXt Xt dBt Xt dBt θ 0 Xt2 dt 0 0 ˆ θT − θ = − R T −θ = RT − RT − θ = − R0 T . (2.3) 2 2 2 2 0 Xt dt 0 Xt dt 0 Xt dt 0 Xt dt RT RT Now 0 Xt dBt is a martingale with the bracket 0 Xt2 dt. Roughly speaking when q RT 2 R RT 1 T 2 T → ∞, 0 Xt dBt is the order of 0 Xt dt. But when T → ∞, T 0 Xt dt → 2 E (X∞ ). So

RT Xt dBt R0 T 2 Xt dt 0

converges to 0 with the order

√1 T

. This argument also

works if Bt is replaced by a square integrable martingale. Unfortunately the stable process Zt is not square integrable. So this naive least squares technique does not apply directly to our case. Another approach is the maximum likelihood ratio. In the Brownian motion case, both the maximum likelihood ratio method and the least squares method give the same estimator. But in our stable process case there is no direct extension of the Girsanov theorem because of the infinite variance property. So the maximum likelihood ratio method cannot be directly applied here neither. 3. Ergodic Case In this section we consider the consistency and the asymptotic distribution of the weighted TFE in ergodic case. That means we assume θ > 0. Then, the solution of the SDE (1.3) can be written in the following way: Z t −θt Xt = e x + e−θ(t−s) dLs 0 Z t Z t −θt −θ(t−s) = e x+m e ds + σ e−θ(t−s) dZs . (3.1) 0

0

The general properties of generalized Ornstein-Uhlenbeck processes driven by Lévy processes have been comprehensively studied in the monograph of Sato [11]. We shall use some important results in Sato [11] freely in this paper. Lemma 3.1. The generalized Ornstein-Uhlenbeck processes {Xt , t ≥ 0} (generated by the triplet (0, ρ, λ)) have a unique invariant distribution µ∞ which is self-decomposable and generated by the triplet (0, ν, γ) with Z Z ∞ 1 ν(B) = ρ(dy) 1B (e−s y)ds, B ∈ B(R) θ R 0 and

γ=

λ 1 + θ θ

Z

|y|>1

y ρ(dy). |y|

Proof. By Theorem 17.5 of Sato [11], we only need to verify that the Lévy measure ρ satisfies the following condition Z log |x|ρ(dx) < ∞. (3.2) |x|>2

180


In fact, we have Z log |x|ρ(dx) |x|>2

c2 log |x| 1(0,∞) (x) + 1+α 1(−∞,0) (x) dx = x1+α |x| |x|>2 Z ∞ Z −2 log x log(−x) = c1 dx + c2 dx 1+α 1+α x (−x) 2 −∞ Z ∞ = (c1 + c2 ) log x · x−1−α dx 2 1 c1 + c 2 log 2 + < ∞. = α2α α Z

c1

This completes the proof.

We can easily find that Xt converges weakly to a random variable Z ∞ m +σ e−θs dZs . X∞ = θ 0 R∞ Note that {Zt } is a Ft -martingale and the random variable 0 e−θs dZs has mean zero. Hence X∞ is a α-stable random variable with mean E[X∞ ] = m θ . By Lemma 3.1 and ergodic theorem, we have Z m 1 T Xt dt = E[X∞ ] = . lim (3.3) T →∞ T 0 θ Remark 3.2. When we don’t specify the way of convergence we always mean the almost sure convergence. We need the following well-known integral version of the Toeplitz Lemma (see Dietz and Kutoyants [2]): Lemma 3.3. If ϕT is a probability measure defined on [0, ∞) such that ϕT ([0, T ]) = 1 and ϕT ([0, K]) → 0 as T → ∞ for each K > 0, then Z ∞ ft ϕT (dt) = f∞ lim T →∞

0

for every bounded and measurable function f : [0, ∞) → R for which the limit f∞ := limt→∞ ft exists. Denote γ(T ) =

Z

T 0

t2 wt2 dt.

We assume that the weight function wt is given so that γ(T ) is well defined for all T > 0 and γ(T ) → ∞ as T → ∞. Theorem 3.4. Assume that θ > 0 and m 6= 0. Then the weighted TFE is strongly consistent in the following sense: lim θˆT = θ

T →∞

P − a.s.


181

Proof. Since the observed process {Xt } is the solution of (1.1), we find RT σ 0 wt2 At Zt dt ˆ θT = θ − R T . 2 2 0 wt At dt

(3.4)

By the strong law of large numbers, Lemma 3.1 and the ergodic theorem, we have lim T −1 ZT = 0

(3.5)

T →∞

(since E[Z1 ] = 0) and 1 T →∞ T

lim T −1 AT = lim

T →∞

Z

T 0

Xt dt = E[X∞ ] =

m . θ

By the Toeplitz Lemma 3.3, we find lim

T →∞

σ

RT

0 RT 0

wt2 At Zt dt wt2 A2t dt

=

lim

T →∞

σ

RT

At t

Zt t

w 2 t2 t

γ(T ) R T A2t wt2 t2 t2 γ(T ) dt 0

0

dt

= 0.

(3.6)

R T wt2 t2 −1 ZT = 0, The last identity follows from the fact that 0 γ(T ) dt = 1, limT →∞ T m −1 and limT →∞ T AT = θ . This proves the theorem. It is well known that the least squares estimator or the maximum likelihood estimator for diffusion processes driven by Brownian motion are asymptotically 1 normal with the order of convergence T − 2 . Here we shall prove that the weighted 1 TFE is asymptotically α-stable with the order of convergence T −(1− α ) . When α is formally set to 2, our result coincides with the classical one. α RT R T R T For notational simplicity, we denote ξ(T ) = 0 twt2 dt τ (T ) = 0 t sws2 ds dt. We assume that the weight function wt satisfies the following condition:

(C1) ξ(T ) → ∞ as T → ∞, ξ(K)/ξ(T ) → 0 as T → ∞ for each K > 0, and ξ(T )T 1/α = O(1). τ (T )1/α

(3.7)

Sufficient conditions on wt satisfying (C1) will be provided later on. Theorem 3.5. If the generalized Ornstein-Uhlenbeck process is ergodic with θ > 0 and m 6= 0, then the weighted TFE is asymptotically α-stable under the condition (C1), i.e. γ(T ) ˆ σθ θ − θ ⇒− κ (3.8) T m τ (T )1/α as T → ∞, where κ is a α-stable random variable with distribution Sα (1, β, 0).

182


Proof. By (3.4), it follows that RT 2 σ wt At Zt dt θˆT − θ = − R0T 2 2 0 wt At dt i h 2 R R T At mσ T 2 −σ 0 t − m θ wt Zt tdt − θ 0 wt Zt tdt = R T At 2 · wt2 t2 dt t 0 =

φ1 (T ) + φ2 (T ) . φ3 (T )

Then, we have τ (T )−1/α φ (T ) + τ (T )−1/α φ (T ) γ(T ) ˆ 1 2 = θ − θ . T 0 γ −1 (T )φ3 (T ) τ (T )1/α

(3.9)

By the ergodic property and the Toeplitz Lemma 3.3, we find Z T 2 m 2 At φ3 (T ) w 2 t2 lim = lim · t dt = P − a.s. T →∞ γ(T ) T →∞ 0 t γ(T ) θ

(3.10)

Now, we consider τ (T )−1/α φ1 (T ). Note that

|τ (T )−1/α φ1 (T )| ≤ στ (T )−1/α ZT∗

Z

T 0

where ZT∗ = sup |Zt |.

At m 2 − tw dt, t θ t

(3.11)

0≤t≤T

RT Denote φ˜1 (T ) = 0 Att − 3.3, it follows that RT :=

2 m θ twt dt. By ergodic property and the Toeplitz Lemma

|φ˜1 (T )| = ξ(T )

Z

T

0

At m twt2 − dt → 0 P − a.s. t θ ξ(T )

as T → ∞. Then, for ε > 0 and δ > 0, we have P(|τ (T )−1/α φ1 (T )| ≥ ε) ≤ P στ (T )−1/α ξ(T )ZT∗ RT ≥ ε

1

≤ P(RT ≥ δ) + P By Markov inequality, we find ! 1 ετ (T ) α ∗ P ZT ≥ ≤ σδξ(T ) ≤

ZT∗

ετ (T ) α ≥ σδξ(T )

!

.

(3.12)

σδξ(T )E[ZT∗ ] 1

τ (T ) α ε σδξ(T ) 1 α

τ (T ) ε

·C

Z

T

dt 0

! α1

=

Cσδξ(T )T 1/α , (3.13) ετ (T )1/α


183

where C is a positive constant depending only on α. By the condition (C1), for any fixed ε > 0, we can choose δ arbitrarily small so that ! 1 ετ (T ) α ∗ lim P ZT ≥ = 0. (3.14) T →∞ σδξ(T ) Obviously, P(RT ≥ δ) → 0 as T → ∞. Therefore, we have lim P(|τ (T )−1/α φ1 (T )| ≥ ε) = 0.

(3.15)

T →∞

Next, we turn to consider the limiting behavior of τ (T )−1/α φ2 (T ). By integration by parts, it follows that # Z T Z T "Z T 2 2 Zt twt dt = sws ds dZt . 0

0

t

By the inner clock property for the α-stable stochastic integral (see Rosinski andh Woyczynski easily that i [9], Kallenberg [5], and Zanzotto0 [12]), we know RT RT 0 2 sws ds dZt has the same distribution as Zτ (T ) , where Z has the same 0 t 0

law as Z. Thus, Zτ (T ) has a α-stable distribution Sα ((τ (T ))1/α , β, 0). By basic properties of α-stable random variable (see Janicki and Weron [4]), we ) converges weakly to a stable distribution Sα ( σ|m| find that τφ(T2 (T θ , sgn(−m)β, 0). )1/α Summarizing we have that γ(T ) ˆ θT − θ 1/α τ (T )

converges weakly to − σθ m κ, where κ is a random variable with α-stable distribution Sα (1, β, 0). Remark 3.6. We shall specify some conditions on wt so that the condition (C1) is satisfied. Of course, we always assume that wt is continuous on [0, ∞). We further assume that there exists a nonnegative function g(s) such that 2 wsT 2 = g(s), ∀s ∈ [0, 1]. T →∞ wT

(3.16)

lim

and s2 g(s) ≤ 1. Now, we can verify that (C1) holds if (3.16) is satisfied. Indeed, we have ξ(T )α T T →∞ τ (T ) lim

=

lim R T

ξ(T )α T

(ξ(T ) − ξ(t))α dt T α = lim R ξ(t) T →∞ T dt 1 − ξ(T 0 )

=

T →∞

0

lim R 1 0 1−

T →∞

1

ξ(sT ) ξ(T )

α

.

ds

(3.17)

184


Also, by L’Hospital’s rule, it follows that for each s ∈ [0, 1] R sT rwr2 dr ξ(sT ) lim = lim R0T T →∞ ξ(T ) T →∞ rwr2 dr 0

2 2 sT wsT s wsT 2 2 = lim = s lim 2 2 = s g(s). T →∞ T wT T →∞ wT

(3.18)

Therefore, we have ξ(T )α T 1 , = R1 2 α T →∞ τ (T ) 0 (1 − s g(s)) ds lim

(3.19)

which implies that the condition (C1) is satisfied. We provide two classes of functions which satisfy the given condition. w2 = s2p . Thus, (i) Let wt = tp . Then, it is seen that limT →∞ wsT 2 T

α

lim

T →∞

1 ξ(T ) T . = R1 2+2p )α ds τ (T ) (1 − s 0

Here, we need to assume that 2 + 2p > 0 or p > −1. By some basic calculation, we find Z 1 Z 1 1 1 1 1 (1 − s2+2p )α ds = (1 − z)α z 2+2p −1 dz = B( , 1 + α). 2 + 2p 2 + 2p 2 + 2p 0 0 So,

ξ(T )T 1/α lim T →∞ τ (T )1/α

=

ξ α (T )T lim T →∞ τ (T )

1/α

=

2 + 2p B(1/(2 + 2p), 1 + α)

(ii) Let wt = eqt , q > 0. Then, it is easy to see that 2 wsT 0, if s ∈ [0, 1) lim 2 = g(s) = 1, if s = 1. T →∞ wT

1/α

.

So, ξ α (T )T 1 = 1. = R1 2 α T →∞ τ (T ) 0 (1 − s g(s)) ds lim

Remark 3.7. Suppose that the condition (C1) is satisfied. As in Remark 3.6, we still assume that there exists a nonnegative function g(s) such that 2 wsT 2 = g(s), ∀s ∈ [0, 1]. T →∞ wT

lim

(3.20)

and s2 g(s) ≤ 1. Under this condition, as shown in Remark 3.6, we have ξ(sT ) = s2 g(s). T →∞ ξ(T ) lim

We claim that γ(T ) 1 = O(T 1− α ). τ 1/α (T )

(3.21)


185

This is equivalent to γ(T ) 1 τ 1/α (T )T 1− α

= O(1)

or γ α (T ) = O(1). τ (T )T α−1 By some basic calculation, we find R

α

T 0

tdξ(t)

α

γ (T ) = lim R T T →∞ τ (T )T α−1 (ξ(T ) − ξ(t))α dt · T α−1 0 R α R α 1 T (ξ(T ) − ξ(sT ))T ds (ξ(T ) − ξ(t))dt 0 0 = lim R 1 = lim R T α α−1 α α−1 T →∞ T →∞ (ξ(T ) − ξ(t)) dt · T 0 (ξ(T ) − ξ(sT )) T ds · T α 0R α R 1 1 2 0 (1 − ξ(sT )/ξ(T ))ds 0 (1 − s g(s))ds = lim R 1 = R1 . (3.22) α 2 α T →∞ 0 (1 − ξ(sT )/ξ(T )) ds 0 (1 − s g(s)) ds lim

T →∞

As in Remark 3.6, we consider the following two special classes of weight functions: (i) Let wt = tp . Then, g(s) = s2p with p > −1. It is easy to find lim

T →∞

γ(T ) = 1/α τ (T )T 1−1/α

2 + 2p 3 + 2p

1 B(1/(2 + 2p), 1 + α) 2 + 2p

−1/α

.

(3.23)

It is not hard to verify that the limit function of p in the right hand side of (3.23) is increasing and is bounded by constant 1. (ii) When wt = eqt , q > 0. It is known from Remark 3.6 that 0, if s ∈ [0, 1), g(s) = 1, if s = 1. It follows that γ(T )

lim

T →∞ τ 1/α (T )T 1−1/α

= 1.

(3.24)

Remark 3.8. The convergence result in Theorem 3.5 means that there is a stable random variable Ξ such that τ (T )1/α Ξ. θˆT − θ ≈ γ(T ) Thus the leading coefficient is proportional to p > −1, it is easy to see that for all T > 0 τ 1/α (T )T 1−1/α = γ(T )

3 + 2p 2 + 2p

τ (T )1/α γ(T ) .

In fact, when wt = tp with

1 B(1/(2 + 2p), 1 + α) 2 + 2p

1/α

=: f (p).

186


4. Non-Ergodic Case In this section, we consider the non-ergodic case, i.e., θ < 0. The solution of the SDE (1.1) is given by Z t Z t θs −θt −θt −θt eθs dZs . (4.1) e ds + σe Xt = e X0 + me Let ηt = X0 + m Let ξt =

Rt 0

Rt 0

θs

e ds + σ

Rt 0

0

0

θs

e dZs . Then, eθt Xt = ηt .

(4.2)

eθs dZs . Then, {ξt }t≥0 is a Lp -bounded cadlag Ft -martingale (1 < p < 1/α

α). Moreover, ξt is a α-stable random variable with distribution Sα (τt , β, 0), where Z t 1 τt = |eθs |α ds = eαθt − 1 . αθ 0 Letting t → ∞, we find that ξt converges to a α-stable random variable with distribution Sα ((−αθ)−1/α , β, 0). Therefore, by martingale convergence theorem, it follows that Z ∞ m +σ eθs dZs := η∞ , P − a.s. (4.3) lim eθt Xt = X0 − t→∞ θ 0 Similar to Section 3, we can define the weighted TFE of θ as follows RT 2 w (Xt − X0 − mt)At dt θˆT = 0 t R T , 2 A2 dt w t t 0 Rt where At = 0 Xs ds. We also have RT σ 0 wt2 At Zt dt ˆ θT − θ = − R T . 2 2 0 wt At dt Denote

h1 (T ) =

Z

T

0

wt2 e−2θt dt and h2 (T ) =

Z

T

0

wt2 e−θt dt .

In this section, we always assume that the weight function wt is given so that hi (T ) → ∞ and hi (K)/hi (T ) → 0 as T → ∞ for each K > 0 and i = 1, 2. We first prove the consistency of the weighted TFE. Theorem 4.1. If θ < 0, then lim (θˆT − θ) = 0

T →∞

P − a.s.

Proof. By the Toeplitz Lemma 3.3, it follows that R t θs Rt −θs ds + η∞ 0 e−θs ds At 0 (e Xs − η∞ )e = lim lim t→∞ t→∞ e−θt e−θt R t −θs Z t e ds λ(t) e−θs θs = ds + η∞ 0 −θt (e Xs − η∞ ) e−θt 0 λ(t) e η∞ , P − a.s., = −θ

(4.4)

(4.5)


187

Rt

e−θ0 s ds. Then, by the Toeplitz Lemma 3.3 again, we have RT 2 σ w At Zt dt lim (θˆT − θ) = lim − R0T t T →∞ T →∞ wt2 A2t dt 0 R T At wt2 e−2θt t σ 0 e−θt · eZ −θt h1 (T ) dt = lim − R 2 −2θt 2 T wt e At T →∞ h1 (T ) dt e−θt 0

where λ(t) =

0

= 0,

since limt→∞ Zt /e−θt = 0,

P − a.s.

(4.6)

P − a.s. (from (3.5)). This completes the proof.

Next, we are going to discuss the asymptotic distribution of θˆT . We assume that the weight function wt satisfies the following condition: (C2) There exist constants C0 > 0, b < 0 such that when T is large enough, wt2 e−θt ≤ C0 eb(T −t) , ∀t ∈ [0, T ]. h2 (T )

(4.7)

We have the following result: Theorem 4.2. If θ < 0 and (C2) holds, then ζ h1 (T ) (θˆT − θ) ⇒ −σ|θ| , 1/α η∞ h2 (T )T

(4.8)

where ζ is a random variable with α-stable distribution Sα (1, β, 0) independent of η∞ . Proof. We have h1 (T ) (θˆT − θ) h2 (T )T 1/α 1 RT −α 2 σh−1 2 (T )T 0 wt At Zt dt =− R T 2 2 h−1 1 (T ) 0 wt At dt

1

−α (−σ)h−1 ZT 2 (T )T = 2 2 2 ηT h−1 1 (T ) 0 wt At dt ! R 1 T −α 2 σh−1 2 (T )T 0 wt At (ZT − Zt )dt + ηT2

ηT2 RT

:= FT (GT + HT ).

RT 0

wt2 At dt

(4.9)

From the Toeplitz Lemma 3.3, it follows that 2 2 −2θt Z T Z T At wt e η2 2 dt = ∞2 . P − a.s. (4.10) lim h−1 (T ) w A dt = lim t t 1 −θt T →∞ T →∞ 0 e h1 (T ) |θ| 0 Thus, we have lim FT = |θ|2

T →∞

P − a.s.

(4.11)

188


Now let us consider GT . Note that (−σ)h−1 2 (T ) GT = ηT

RT 0

1

wt2 At dt T − α ZT · . ηT

By the Toeplitz Lemma 3.3 again, we get 2 −θt Z T Z T At wt e η∞ 2 lim h−1 (T ) w A dt = lim dt = t t 2 −θt T →∞ T →∞ e h (T ) |θ| 2 0 0

(4.12)

(4.13)

almost surely. Consequently (−σ)h−1 2 (T ) lim T →∞ ηT

RT

wt2 At dt

0

=−

σ |θ|

P − a.s.

(4.14)

For the second factor in GT , we have 1

1

1 T − α (ZT − Z α1 ) + T − α Z α1 T − α ZT T T = ηT η α1 + (ηT − η α1 )

We have the following claims. 1 (1) The random variable T − α (ZT − Z 1 α −1

1/α

(4.15)

T

T

T

1 α

) has a α-stable distribution Sα (σ(1 −

T ) , β, 0), which converges weakly to a random variable ζ with stable distribution Sα (1, β, 0) as T → ∞. (2) By strong law of large numbers, we have 1

lim T − α Z

T →∞

T

1 α

= 0 P − a.s.

(3) It is clear that lim η

1

T →∞ T

= η∞

1 α

P − a.s.

(4) T − α (ZT − Z α1 ) and η α1 are independent. T T (5) We have that ηT − η α1 converges to zero in probability as T → ∞. T Proof of (5). By the definition of ηt , we find Z T Z T ηT − η α1 = m 1 eθs ds + eθs dZs . 1 T

It follows that

T

T

α

α

Z T θs |ηT − η α1 | ≤ |m| 1 e ds + 1 e dZs T α α T T Z 1 T eθT − eθT α θs ≤ |m| · + 1 e dZs . Tα θ Z

T

θs

(4.16)

The first term on the right hand side of (4.16) converges to zero as T → ∞. The second term converges to zero in probability as T → ∞, since R ( Z ) T 1 θs T e dZ E s Tα θs P 1 e dZs > ε ≤ Tα ε


C ≤ ε

Z

T T

1 α

eαθs ds

! α1

1

eαθT − eαθT α αθ

C ≤ ε

! α1

189

,

(4.17)

which tends to zero as T → ∞ for any given ε > 0 and some constant C > 0. From all the claims (1)-(5), we conclude that 1

T − α ZT ζ ⇒ ηT η∞

(4.18)

where ζ and η∞ are independent. Combining (4.14) and (4.18), we find GT ⇒ −

σ ζ |θ| η∞

(4.19)

as T → ∞. Finally, we shall prove that HT → 0 in probability as T → ∞. We have Z T − α1 −1 2 h2 (T ) wt At (ZT − Zt )dt σT 0 Z t Z T 1 wt2 Xs ds |ZT − Zt |dt ≤ σT − α h−1 2 (T ) 0 0 Z T Z t 1 − α −1 θs −θs ≤ σT h2 (T ) |e Xs |e ds |ZT − Zt |wt2 dt 0

0

1 σ sup |eθt Xt | · T − α h−1 ≤ 2 (T ) |θ| t≥0

Z

T

0

|ZT − Zt |wt2 e−θt dt.

(4.20)

It is easy to see that supt≥0 |eθt Xt | is almost surely finite. We claim that the last RT 1 2 −θt dt in the above inequality converges to zero factor T − α h−1 2 (T ) 0 |ZT − Zt |wt e in probability. Indeed, we have for T large enough # " Z T

1

E T − α h−1 2 (T )

≤T

1 −α

≤ C0 T

Z

0

|ZT − Zt |wt2 e−θt dt

T

E[|ZT − Zt |] 0

1 −α

1

= C0 T − α

Z Z

T

wt2 e−θt dt h2 (T ) 1

C(1, α)(T − t) α eb(T −t) dt

0 T

1

C(1, α)u α ebu du 0 Z ∞ 1 1 ≤ C(1, α)C0 T − α u α ebu du 0

≤ C(1, α)C0 T

1 −α

Γ(1 +

1 1 )|b|−(1+ α ) , α

(4.21)

4Γ(−1/α) which tends to zero as T → ∞, where C(1, α) = α√ (see Zolotarev [13]). πΓ(−1/2) R 1 T 2 This implies that σT − α h−1 2 (T ) 0 wt At (ZT − Zt )dt converges to zero in probability as T → ∞. So, HT → 0 in probability as T → ∞. From (4.9), (4.11), (4.19),

190


(4.20) and (4.21), we conclude that h1 (T ) 1

h2 (T )T α

(θˆT − θ) ⇒ −σ|θ|

ζ , η∞

(4.22)

where ζ is a Sα (1, β, 0) random variable independent of η∞ . This completes the proof. Remark 4.3. We consider two special classes of weight functions. (i) Let wt = tp , p ≥ 0. Some basic calculation yields h2 (T ) =

Z

T 0

wt2 e−θt dt =

Z

T

t2p e−θt dt ≥ C1 T 2p e−θT ,

(4.23)

0

for each T ≥ T0 with some T0 > 0 and some C1 > 0. It follows that for T large enough t2p e−θt 1 θ(T −t) wt2 e−θt ≤ ≤ e , t ∈ [0, T ]. 2p −θT h2 (T ) C1 T e C1

(4.24)

This implies that (C2) is satisfied with b = θ. (ii) Let wt2 = ert . Then, h2 (T ) =

Z

T

ert e−θt dt =

0

e(r−θ)T − 1 . r−θ

(4.25)

We have wt2 e−θt ≤ C2 e(θ−r)(T −t) , h2 (T )

(4.26)

for T large enough and b = θ − r < 0. Thus, if r > θ, then the condition (C2) is satisfied. Remark 4.4. We claim that h1 (T ) = O(e−θT T −1/α ). h2 (T )T 1/α

(4.27)

This is equivalent to h1 (T ) = O(e−θT ). h2 (T ) We assume that wt is a continuously differentiable function satisfying 0

w lim T = a, T →∞ wT

(4.28)


191

where a is a constant such that −2θ+2a > 0 and 2a−θ > 0. Then, by L’Hospital’s rule, we have h1 (T )eθT T →∞ h2 (T ) lim

wT2 e−2θT eθT + h1 (T )eθT θ T →∞ wT2 e−θT h1 (T ) = 1 + θ lim 2 −2θT T →∞ wT e

=

lim

wT2 e−2θT 0 2 −2θT T →∞ w e (−2θ) + 2wT wT e−2θT T

= 1 + θ lim = 1+

θ 2a − θ = . 2(a − θ) 2(a − θ)

(4.29)

Two special cases: (i) Let wt = tp , p ≥ 0. Then, 0

lim

T →∞

wT pT p−1 = lim = 0. T →∞ wT Tp

So, h1 (T )eθT = 1/2. T →∞ h2 (T ) lim

(4.30)

(ii) Let wt = ert/2 , r > θ. Then, r rT /2 e wT r = lim 2 rT /2 = . T →∞ wT T →∞ e 2 0

lim

So, h1 (T )eθT r−θ = . T →∞ h2 (T ) r − 2θ lim

(4.31)

References 1. Dietz, H. M.: Asymptotic behavior of trajectory fitting estimators for certain non-ergodic SDE; Statistical Inference for Stochastic Processes 4 (2001) 249–258. 2. Dietz, H. M. and Kutoyants, Yu. A.: A class of minimum-distance estimators for diffusion processes with ergodic properties; Statistics and Decisions 15 (1997) 211–227. 3. Diets, H. M. and Kutoyants, Yu. A.: Parameter estimation for some non-recurrent solutions of SDE; Statistics and Decisions 21 (2003) 29–45. 4. Janicki, A. and Weron, A.: Simulation and Chaotic Behavior of α-stable Stochastic Processes. Marcel Dekker, 1994. 5. Kallenberg, O.: Some time change representations of stable integrals, via predictable transformations of local martingales; Stochastic Process. Appl. 40 (1992) 199–223. 6. Kutoyants, Yu. A.: Minimum distance parameter estimation for diffusion type observations; C. R. Acad. Sci. Paris, Serie I, 312 (1991) 637–642. 7. Kutoyants, Yu. A.: Statistical Inference for Ergodic Diffusion Processes. Springer-Verlag, London, Berlin, Heidelberg, 2004. 8. Liptser, R. S. and Shiryaev, A. N.: Statistics of Random Processes: II Applications. Second Edition, Applications of Mathematics, Springer-Verlag, Berlin, Heidelberg, New York, 2001. 9. Rosinski, J. and Woyczynski, W. A.: On Itˆ o stochastic integration with respect to p-stable motion: inner clock, integrability of sample paths, double and multiple integrals; Ann. Probab. 14 (1986) 271–286. 10. Samorodnitsky, G. and Taqqu, M. S.: Stable non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, New York, London, 1994.

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11. Sato, K. I.: L´ evy Processes and Infinitely Divisible Distributions. Cambridge University Press, 1999. 12. Zanzotto, P. A.: Representation of a class of semimartingales as stable integrals; Theory Probab. Appl. 43 (1998) 666–676. 13. Zolotarev, V. M.: One-dimensional Stable Distributions. American Mathematical Society, 1986. Yaozhong Hu: Department of Mathematics, University of Kansas, Lawrence, Kansas E-mail address: [email protected] URL: http://www.math.ku.edu/∼hu Hongwei Long: Department of Mathematical Sciences, Florida Atlantic University, Florida E-mail address: [email protected] URL: www.math.fau.edu/Long/hlong.htm


GENERATORS OF DYNAMICAL SYSTEMS ON HILBERT MODULES GHOLAMREZA ABBASPOUR TABADKAN AND MICHAEL SKEIDE Abstract. We characterize the generators of dynamical systems on Hilbert modules as those generators of one-parameter groups of Banach space isometries which are ternary derivations. We investigate in how far a similar condition can be expressed in terms of generalized derivations.

1. Introduction Let E be a Hilbert module over a C ∗ –algebra B. A generalized unitary on E is a surjection u on E that fulfills hux, uyi = ϕ(hx, yi),

x, y ∈ E

(GU)

for some automorphism ϕ of B. We will also say u is a ϕ–unitary . A generalized derivation of E is a densely defined linear map δ : E ⊃ dom(δ) → E that fulfills δ(xb) = δ(x)b + xd(b),

x ∈ dom(δ), b ∈ dom(d)

(GD)

for some derivation d : B ⊃ dom(d) → B of B, in such a way that dom(δ) is a right dom(d)–module. We will also say δ is a d–derivation. A dynamical system on a Hilbert B–module E is a strongly continuous one-parameter group u = ut t∈R of generalized unitaries. Abbaspour, Moslehian and Niknam [2, 1] defined dynamical systems on Hilbert modules and started the investigation of their generators. They showed that the generator of a dynamical system on a full Hilbert B–module is a generalized derivation (of course, with respect to the generator of the associated C ∗ –dynamical system ϕ = ϕt t∈R on B such that every ut is a ϕt –unitary). However, a bounded generalized derivation on E with respect to a bounded ∗–derivation on B need not generate a dynamical system on E; see Example 3.14. Even if we require that a closed and densely defined linear map δ generates a group of Banach space isometries on E, the condition that δ be a generalized derivation with respect to a ∗–derivation d of B turns out to be sufficient, only under algebraic and analytic conditions on the domain of d; see Theorems 3.12 and 3.18. These conditions depend manifestly on the domain of δ and, therefore, cannot be understood in terms of d alone. 2000 Mathematics Subject Classification. Primary 46L55; Secondary 46L53, 46L08. Key words and phrases. Quantum dynamics, quantum probability, Hilbert modules, dynamical systems. GAT is supported by a grant from Minstry of sience, Reaserch and Technology of Iran (MSRT). MS is supported by research funds of the University of Molise and the Italian MIUR (PRIN 2005). 193

194

GHOLAMREZA ABBASPOUR TABADKAN AND MICHAEL SKEIDE

It is the main scope of these notes to find a better algebraic condition that depends only on δ. This condition will be in terms of ternary maps. Ternary maps have the advantage that they their definition refers exclusively to the module E, not to the C ∗ –algebra B. A ternary automorphism of E is a bijection u on E that fulfills u(xhy, zi) = (ux)huy, uzi,

x, y, z ∈ E.

(TU)

In Section 2 we show that the generalized isometries from a full Hilbert B–module E to a Hilbert C–module F are exactly the ternary homomorphims. As a special case this includes the statement that the generalized unitaries on a full Hilbert B–module are exactly its ternary automorphisms. This frees the discussion from worrying about existence of an automorphism ϕ of B. In fact, the main problem in the proof is to show that existence of such an automorphism is automatic. Consequently, the dynamical systems on a full Hilbert module E are exactly the strongly continuous one-parameter groups of ternary automorphisms. A ternary derivation of E is a densely defined linear map δ : E ⊃ dom(δ) → E that fulfills δ(xhy, zi) = δ(x)hy, zi + xhδ(y), zi + xhy, δ(z)i, x, y, z ∈ E (TD)

where dom(δ) dom(δ) , dom(δ) ⊂ dom(δ), that is, dom(δ) is invariant under the ternary product (x, y, z) 7→ xhy, zi. In Section 3 we show that every ternary derivation of a full Hilbert module is a generalized derivation, while the converse fails. Generators of dynamical systems are always ternary derivations. We show also a sort of converse: If a linear densely defined map on E is the generator of a C0 –group (that is, a strongly continuous one-parameter group of Banach space isometries) on E, then this group is a dynamical system, if and only if δ is a ternary derivation. This reduces the problem of characterizing the generators to the wellknown general analytic criteria based on Hille-Yosida theory that state when δ is the generator of a C0 –group, and the purely algebraic question whether δ is a ternary derivation. We see that we have a satisfactory description of generators of dynamical systems on Hilbert modules in terms of ternary derivations, while the larger part of Section 3 illustrates that similar statements in terms of generalized derivations are possible only under rather hard analytical hypothesis. We note, too, that the condition that the Hilbert B–module E be full is not critical as long as we speak about ternary maps. Restrictions that arise in the case of generalized unitaries on a Hilbert module E when B is not chosen minimal have been analyzed in Skeide [8]. The scope of these notes is to characterize the generators of dynamical systems on Hilbert modules, and that scope does not depend on potential applications. We prefer to outline a view of these potential applications mainly for stochastic analysis, in particular for stochastic analysis in quantum probability, in Section 4. Conventions and notations. A pre-Hilbert B–module is a right module E over a (pre-)C ∗ –algebra, with a sesquilinear inner product h•, •i : E × E → B that satisfies hx, ybi = hx, yib (x, y ∈ E; b ∈ B), hx, xi ≥ 0 and hx, xi = 0 ⇒ x = 0 (x ∈ E). A Hilbert B–module is a pre-Hilbert B–module that is complete in

GENERATORS OF DYNAMICAL SYSTEMS ON HILBERT MODULES

195

p the norm kxk := hx, xi. A pre-Hilbert B–module E is full , if the range ideal BE := spanhE, Ei is dense in B. By Ba (E) and K(E) we denote the C ∗ –algebras of all adjointable operators and of all compact operators, respectively, on E, where K(E) is the completion of the pre-C ∗ –algebra F(E) of finite-rank operators which is spanned by the rank-one operators xy ∗ : z 7→ xhy, zi. 2. Generalized isometries versus ternary homomorphisms Unitaries on or between Hilbert modules are inner product preserving surjections. For isometries, surjectivity is missing. For generalized unitaries on a Hilbert module in (GU) the condition that the surjection preserves inner products is modified to that it preserves inner products up to a fixed automorphism of the algebra. When the unitary is between different Hilbert modules, it is not even necessary that these are modules over the same algebra. In this section we investigate generalized isometries between Hilbert modules. Let E be a Hilbert B–module and let F be a Hilbert C–module. A generalized isometry from E to F is a map u : E → F that fulfills (GU) for some homomorphism ϕ : B → C. We will also say u is a ϕ–isometry . Calculating the norm of ux + (uy)ϕ(b) − u(x + yb), we find that every ϕ–isometry u is ϕ–linear , that is, ux + (uy)ϕ(b) = u(x + yb) (x, y ∈ E; b ∈ B). Inserting scalar multiples of an approximate unit, we see that ϕ–linearity implies C–linearity. Obviously, a ϕ–isometry has norm 1, unless ϕ BE is 0. A ternary homomorphism from E to F is a map u : E → F that fulfills (TU). Obviously, every ϕ–isometry is a ternary homomorphism. It is our scope to show that, at least if E is full, then every ternary homomorphism is also a ϕ–isometry. Theorem 2.1. For a map u from a full Hilbert B–module E to a Hilbert C–module F the following statements are equivalent: (1) u is a generalized isometry. (2) u is a ternary homomorphism. Proof. Given a ternary homomorphism u from a full Hilbert B–module E to a Hilbert C–module F , it is our job to find a homomorphism ϕ : B → C such that u fulfills (GU). First, we observe that for full E such a homomorphism is determined uniquely by (GU). The attempt to define the homomorphism ϕ first on the preC ∗ –algebra BE by hx, yi 7→ hux, uyi and then to show that it is bounded by appealing to Muhly, Skeide and Solel [6, Corollary 1.20] has been put into practise in [1] under the assumption that u is linear. Here we follow a different road. Let us observe that if a suitable ϕ exists, then u must be ϕ–linear. So, necessarily we must have (ux)ϕ(b) = u(xb) for all x ∈ E. We use this property to define a left action ϕ(b) of b ∈ B on the pre-C ∗ –algebra CuE := spanhuE, uEi considered pre-Hilbert module over itself in the usual way, that is, with inner product hc, c0 i = c∗ c0 and right action simply by multiplication. We put ϕ(b)hux, uyi := hu(xb∗ ), uyi

196


and we must show, in a first step, that this well-defines a homomorphism into Ba (CuE ). As, clearly, ϕ(b)ϕ(b0 )hux, uyi = ϕ(bb0 )hux, uyi (so that, once welldefined, ϕ is multiplicative), it is sufficient to show that ϕ(b∗ ) is a formal adjoint of ϕ(b) on the spanning subset of elements of the form hux, uyi. From this follow both that ϕ(b) is well-defined and that ϕ(b∗ ) = ϕ(b)∗ . We start by observing that

c, hux, uyi = c∗ hux, uyi = h(ux)c, uyi for all c ∈ CuE . Using this two times, we find D E D E hux, uyi, ϕ(b)hux0 , uy 0 i = hux, uyi, hu(x0 b∗ ), uy 0 i

= u(x0 b∗ )hux, uyi, uy 0 = u(x0 b∗ hx, yi), uy 0

= u(x0 hxb, yi), uy 0 = (ux0 )hu(xb), uyi), uy 0 D E D E = hu(xb), uyi, hux0 , uy 0 i = ϕ(b∗ )hux, uyi, hux0 , uy 0 i .

Like every homomorphism from a C ∗ –algebra into the adjointable operators on a pre-Hilbert module, ϕ maps into the bounded operators, and like every homomorphism from a C ∗ –algebra into a pre-C ∗ –algebra, ϕ is a contraction. Next we observe that ϕ(hx, yi) acts on the element hux0 , uy 0 i of CuE simply by multiplication from the left with the element hux, uyi. The subalgebra ϕ(BE ) of Ba (CuE ) is nothing but CuE , which, of course, is faithfully contained in Ba (CuE ). In other words, ϕ is the unique continuous extension from BE to B = BE of ϕ BE and, therefore, maps into CuE ⊂ C. Clearly, ϕ turns u into a ϕ–isometry. Corollary 2.2. Every ternary homomorphism is linear and contractive. Proof. The only thing that remains is to remark that if E is not full, then we may simply turn E into a full Hilbert module by restricting to BE . Observation 2.3. Note that a ternary homomorphism is injective, if and only if the homomorphism ϕ : BE → C that turns it into a ϕ–isometry is injective. (Every surjective but noninjective endomorphism of B is an example for a surjective ternary homomorphism that is not injective.) This shows, in particular, that the ϕ induced by a ternary automorphism on a full Hilbert module is itself an automorphism. Corollary 2.4. The group of generalized unitaries on a full Hilbert module E coincides with the group of ternary automorphisms of E. Therefore, the dynamical systems on a Hilbert module E are exactly the C0 –groups of ternary automorphisms. Remark 2.5. By the construction in the proof of Theorem 2.1 every C0 –group u = ut t∈R of ternary automorphisms of a Hilbert B–module comes along with a (unique) family of automorphisms ϕt of BE and, obviously, the ϕt form a C ∗ –dynamical system. These automorphisms ϕt do, in general, not necessarily extend to automorphisms of B; see [8]. Therefore, for not necessarily full E there are, in general, more groups of ternary automorphisms than groups of generalized unitaries. In the general case, it seems, therefore, advisable to define a dynamical system on a Hilbert module as a C0 –group of ternary automorphisms.


197

Remark 2.6. By [8, Observation 1.4] (for instance) we know that every surjective ϕ–isometry from a Hilbert B–module E to a Hilbert C–module F extends to a homomorphism between the extended linking algebras Φ =

„

ϕ u∗ u ϑ

«

:

„

B E∗ E Ba (E)

«

−→

„

C F∗ F Ba (F )

«

that restricts to a homomorphism between the usual linking algebras „

C F∗ F K(F )

«

∗

∗

„

B E∗ E K(E)

«

→

∗

. (Here u (x ) := (ux) , while ϑ(a) acts on y = ux in the only possible way, namely, ϑ(a)(ux) = u(ax). Well-definedness of ϑ(a) follows in a way paralleling the proof of well-definedness of ϕ in the proof of Theorem 2.1.) Therefore, generalized isometries and, consequently, also ternary homomorphisms are even completely contractive. (One may obtain this result also as in [1], by showing that every inflation un of u is a ϕn –isometry from Mn (E) to Mn (F ) and, therefore, a contraction.) This improves on a result on ternary homomorphisms of C ∗ –algebras by Bracic and Moslehian [3]. A ternary homomorphism η from E into the Hilbert B(G)–module B(G, H) for two Hilbert spaces G and H is what has been called a representation of E from G to H in Skeide [7]. The preceding discussion improves also on [7, Theorem A.4] where the extendibility of η to a representation of the linking algebra has been shown under the explicit hypothesis that η be completely bounded. Now we see that this hypothesis is fulfilled automatically.

3. Generalized derivations versus ternary derivations It is easy to see that the generator δ of a dynamical system u = ut t∈R on a full Hilbert B–module E is a generalized derivation; see [2] and cf. also Corollary 3.6. A possible choice for the derivation d in (GD) is simply the generator of the C ∗ –dynamical system ϕ = ϕt t∈R associated with u, that is determined uniquely by the requirement that every ut is a ϕt –unitary; see Remark 2.5. But, even if we know that the generator δ of C0 –group on E is a d–derivation, then it is not possible to conclude that δ generates a dynamical system without making further analytical assumptions about d and algebraic assumptions about the domain of d, see Theorem 3.12. These algebraic conditions are relative to δ, that is, they cannot be formulated intrinsically in terms of the derivation d of B alone, but depend explicitly on δ. On the other hand, it is easy to formulate these conditions intrinsically in terms of δ alone: δ must be a ternary derivation. We study, first, the intrinsic description of the generators of dynamical systems on Hilbert modules as ternary derivations (Theorem 3.1). Then, we explain the relationship between ternary derivations and generalized derivations. We will see that there is a particular derivation dδ (Theorem 3.5) that allows to formulate Theorem 3.1 in terms of generalized derivations (Theorem 3.18). In Theorem 3.19 we summarize all criteria and add one more in terms of the linking algebra. Theorem 3.1. Let u = ut t∈R be C0 –group on a Hilbert B–module E. Then u is a dynamical system if and only if its generator δ is a ternary derivation.

198


Proof. Recall that the generator of a C0 –group u is defined as ut x − x δ(x) := lim t→0 t for all x for which the limit exists. Further, recall that this domain dom(δ) contains a dense core of entire analytic vectors. That means, the subspace A(δ) ⊂ T n n∈N dom(δ ) that consists of all vectors x for which for all t ∈ R the series ∞ n n X t δ x n! n=0

converges absolutely to the limit ut x is dense in E and δ is the closure of δ A(δ). Suppose δ is the generator of a dynamical system u. Let x, y, z ∈ dom(δ). By Corollary 2.4 all ut are ternary automorphisms, so that ut (xhy, zi) − xhy, zi (ut x)hut y, ut zi − xhy, zi = t t Du y − y E D u z −zE ut x − x t t . hut y, ut zi + x , ut z + x y, = t t t As all ut are contractions, the families ut x and ut y are bounded uniformly. So the limit of the preceding expression exists and is equal to δ(x)hy, zi + xhδ(y), zi + xhy, δ(z)i. This shows both that xhy, zi ∈ dom(δ) and that δ is a ternary derivation. Conversely, suppose that δ is a ternary derivation, and choose entire analytic elements x, y, z ∈ A(δ). By a routine induction we show the ternary generalized Leibniz rule X n! k δ (x)hδ ` (y), δ m (z)i. δ n (xhy, zi) = k!`!m! k,`,m∈N0 k+`+m=n

¿From this, one easily concludes that xhy, zi is also in A(δ) and that ut fulfills (TU) on the dense subset A(δ). By contractivity of ut , this extends to all of E so that ut is a ternary automorphism. Remark 3.2. For general results about C0 –groups we refer to Bratteli and Robinson [4]. In particular, the problem to decide, whether a linear densely defined map is the generator of C0 –group, we leave entirely to the comprehensive treatment in [4]. But, once we have such a generator, we see that the problem whether the generated group is a dynamical system, is equivalent to the question whether the generator is a ternary derivation. Thus, we have a complete separation into the general analytic criteria of the Banach space theory that determine when δ is a generator (which we do not treat here) and the completely algebraic criterion in Theorem 3.1. Theorem 3.1, in principle, completely settles the problem to characterize the generators of dynamical systems on Hilbert modules. Fullness, is not at all a critical assumption, because if necessary we may always make B smaller. In the remainder of this section we deal with the problem to find similarly useful criteria in terms of generalized derivations. We start by establishing a connection between ternary derivations and a special sort of generalized derivations on the algebraic


199

level. However, a full correspondence we will obtain only under the assumption that the derivations in question generate C0 –groups. On the level of derivations the assumption of fullness becomes much more vital, as we do not see a possibility to show that the derivation of B that turns a map δ into a d–derivation restricts to a derivation of BE . The following uniqueness result, depending essentially on fullness, is crucial for all other statements that follow. Proposition 3.3. Let δ : E ⊃ dom(δ) → E be a densely defined linear map on a full Hilbert B–module E. Then for every dense subalgebra B0 of B, there is at most one derivation d of B with domain dom(d) = B0 that turns δ into a d–derivation. Corollary 3.4. If d1 , d2 are derivations of B and if δ is a d1 –derivation and a d2 –derivation of a full Hilbert B–module E, then d1 ⊂ d 2

⇐⇒

dom(d1 ) ⊂ dom(d2 ).

Proof of Proposition 3.3. If δ is a d–derivation, then we have xd(b) = δ(xb) − δ(x)b for all x ∈ dom(δ) and all b ∈ dom(d). Since E is full and dom(δ) is dense in E, the preceding equation determines d(b) ∈ B uniquely. Theorem 3.5. Every ternary derivation δ of a full Hilbert B–module E is also a generalized derivation. More precisely, there is a unique derivation d δ of B on the dense domain dom(dδ ) := spanhdom(δ), dom(δ)i that fulfills dδ (hx, yi) = hδ(x), yi + hx, δ(y)i.

(3.1)

dδ turns δ into a dδ –derivation. Moreover, dδ is a ∗–derivation. Proof. Suppose we have a derivation dδ on the given domain, that turns δ into a dδ –derivation. Then (following the proof of Proposition 3.3) for the uniquely determined values of dδ (hx, yi) we find x dδ hy, zi = δ(xhy, zi) − δ(x)hy, zi = x hδ(y), zi + hy, δ(z)i (3.2) for all x, y, z ∈ dom(δ). We see that if a suitable derivation dδ exists, then it must fulfill (3.1). In particular, dδ is necessarily a ∗–derivation. So the only remaining questions are, firstly, whether (3.1) always well-defines a linear map dδ : spanhdom(δ), dom(δ)i −→ B, and, secondly, whether this map is a (necessarily ∗–) derivation. For P the first question, suppose yi , zi are finitely many elements of dom(δ) fulfilling i hyi , zi i = 0. Then X X X hδ(yi ), zi i + hyi , δ(zi )i = δ x hyi , zi i − δ(x) hyi , zi i = 0 x i

i

i

for all x ∈ dom(δ), so that dδ is, indeed, well-defined. For the second question, let us compute the inner product of an element w ∈ dom(δ) with (3.2) and the

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adjoint of the resulting equation. Using this, we find dδ hw, xi hy, zi + hw, xi dδ hy, zi = hδ(w), xi + hw, δ(x)i hy, zi + hw, xi hδ(y), zi + hy, δ(z)i D E = hδ(w), xihy, zi + w , δ(x)hy, zi + xhδ(y), zi + xhy, δ(z)i

= dδ hw, xihy, zi . = δ(w), xhy, zi + w , δ(xhy, zi) = dδ w, xhy, zi By linearity this extends to dδ (b)b0 + bdδ (b0 ) = dδ (bb0 ) for all b, b0 in the domain spanhdom(δ), dom(δ)i. Corollary 3.6. [2] Every generator of a dynamical system on a full Hilbert module is a generalized derivation. Corollary 3.7. Every ternary derivation of a full Hilbert module extends as a ∗–derivation to the linking algebra. Proof. Suppose δ is ternary derivation of the full Hilbert B–module. By Theorem 3.5 this determines the ∗–derivation dδ of B which is the candidate for how to extend δ to the corner B of the linking algebra. To find the extension to K(E), we observe that E ∗ is a full Hilbert K(E)–module, the dual module of E, with inner product hx∗ , y ∗ i := xy ∗ ∈ K(E) and the right action x∗ a := (a∗ x) of elements a ∈ K(E) (or even in Ba (E)). Of course, δ ∗ (x∗ ) := δ(x)∗ defines a ternary derivation of E ∗ with domain dom(δ ∗ ) := dom(δ)∗ , and by Theorem 3.5 there is a unique ∗–derivation dδ∗ of K(E) defined on the domain dom(dδ∗ ) = span dom(δ) dom(δ)∗« , turning δ ∗ into a dδ∗ –derivation. It is routine to check that „ „ ∗ ∗ ∗« b y δ (b) δ (y ) defines a ∗–derivation of the linking algebra. 7→ dδ(x) d ∗ (a) x a δ

If δ is a ternary derivation, the derivation dδ plays a distinguished role as it is related more directly to questions of closability than any other derivation d that turns δ into a d–derivation. The following Proposition 3.9 settles some of these closability questions in the setting of general derivations, while in Theorems 3.12 and 3.18 the assumption that the maps are generators of C0 –groups is crucial. The following task, needed in the proofs of Proposition 3.9(1b) and of Lemma 3.15, is so useful that we prefer to formulate it separately. Lemma 3.8. Suppose the elements of the Hilbert B–module E separate the points of B, that is, xb = 0 for all x ∈ E implies b = 0. (For instance, suppose E is full.) Then kbk = sup kxbk . kxk≤1

Proof. By setting bx∗ := (xb∗ )∗ we define a representation of B by adjointable (and, therefore, bounded; see the proof of Theorem 2.1) operators on the dual module E ∗ (see the proof of Corollary 3.7). By hypothesis, this representation is faithful and, therefore, isometric. In other words, the operator norm of the action of b ∈ B as operator on E ∗ coincides with the norm of b as element of B. Observing that kxbk = k(xb)∗ k, this is precisely the statement of the lemma.


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Proposition 3.9. Let E be a full Hilbert B–module. (1) Let δ be a d–derivation of E. (a) If δ is closable, then so is d. (b) If δ is bounded, then so is d. (2) Let δ be a ternary derivation of E. Then δ is closable, if and only if d δ is closable. (3) Let δ be a ternary derivation and a d–derivation of E. If dδ is closable, then so is d. Proof. (1a) Suppose that δ is a closable d–derivation. Let bn → 0 be a sequence in dom(d) such that d(bn ) → b ∈ B. Then for every x ∈ dom(δ) we find δ(xbn ) = δ(x)bn + xd(bn ) −→ 0 + xb. As xbn → 0 and δ is closable, it follows that δ(xbn ) → 0, so that xb = 0 for all x ∈ dom(δ). As E is full, this implies b = 0. So, d is closable. (1b) Suppose that δ is a bounded d–derivation. By Lemma 3.8, for every b ∈ dom(d) we find an x in the unit ball of E such that kxd(b)k ≥ 21 kd(b)k. So, kd(b)k ≤ 2 kxd(b)k ≤ 2 kδ(xb)k + kδ(x)bk ≤ 4 kδk kbk. (2) Suppose now that δ is a ternary derivation such that dδ is closable. Let xn → 0 be a sequence in dom(δ) such that δ(xn ) → x ∈ E. Then for every y ∈ dom(δ) we find dδ (hy, xn i) = hy, δ(xn )i + hδ(y), xn i −→ hy, xi + 0. As hy, xn i → 0 and dδ is closeable, it follows that dδ (hy, xn i) → 0, so that hy, xi = 0 for all y ∈ dom(δ) and, therefore, x = 0. So, δ is closable. If E is full, then, by Part (1a), also the converse is true. (3) If dδ is closable, then, by (2), δ is closable so that, by (1a), d is closable. Remark 3.10. Boundedness of d is not sufficient for boundedness of δ. In fact, every generator of a unitary C0 –group on a Hilbert module that is not uniformly continuous is an unbounded ternary derivation and a 0–derivation for the trivial derivation 0 : b 7→ 0. Observation 3.11. If, in (3), dom(d) does not contain dom(dδ ), then we may easily replace d by the derivation d0 defined on alg∗ dom(dδ ), dom(d) , the ∗–algebra generated by dom(dδ ) and dom(d), that is determined uniquely (see Proposition 3.3!) by the requirement that δ be a d0 –derivation. (If such a d0 exists, then, again by Proposition 3.3, this implies also that d0 is the unique extension as a derivation of d and dδ to the new domain.) Let us first define d0 on the domain dom(d)∪dom(dδ ) as d0 (b) := d(b) for b ∈ dom(d) and d0 (b) := dδ (b) for b ∈ dom(dδ ). (Once more, by the proof of Proposition 3.3, this is well-defined as d and dδ coincide on the intersection of their domains.) By induction we show that for every choice of elements b1 , . . . , bn from dom(d) ∪ dom(dδ ) and for all x ∈ dom(δ) (so that also xb1 . . . bn is in dom(δ)) δ(xb1 . . . bn ) − δ(x)b1 . . . bn = x d0 (b1 )b2 . . . bn + . . . + b1 . . . bn−1 d0 (bn ) . This shows that for every b in the new domain there is a uniqe b0 ∈ B satisfying xb0 = δ(xb)−δ(x)b and that the map d0 : b 7→ b0 is linear. Clearly, d0 is a derivation

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and δ is a d0 –derivation. By Parts (2) and (3), d0 is closable, if and only if dδ or, equivalently, if δ is closable. In other words, every derivation d that turns a closable ternary derivation δ of a full Hilbert B–module into a d–derivation admits a unique minimal closed extension d0 ⊃ dδ , and δ is also a d0 –derivation. Theorem 3.12. Suppose δ is the generator of a dynamical system u on a full Hilbert B–module E and a d–derivation for some (by Theorem 3.1 and Proposition 3.9(3), necessarily closable) derivation d of B. Denote by d ϕ the generator of the C ∗ –dynamical system ϕ associated with u. (1) The unique minimal closed extension d0 ⊃ dδ of d (see Observation 3.11) is the generator of ϕ, if and only if d ⊂ dϕ . (2) If d ⊂ dϕ , then for d = dϕ it is necessary and sufficient that d ⊃ dδ . Proof. As in the proof of Theorem 3.1 we see that dom(dδ ) ⊂ dom(dϕ ) and that the span of hA(δ), A(δ)i is a dense subspace of entire analytic elements of dom(d ϕ ). Therefore, every subspace D with hA(δ), A(δ)i ⊂ D ⊂ dom(dϕ ) is a core for dϕ . In particular, dom(dδ ) is a core for dϕ . (1) If d 6⊂ dϕ , then d ⊂ d0 ⊂ d0 6⊂ dϕ . Conversely, if d ⊂ dϕ then also d0 ⊂ dϕ (because dδ ⊂ dϕ and, therefore, alg∗ (dom(d), dom(dδ )) ⊂ dom(dϕ )), so that d0 ⊂ dϕ = dϕ . (2) If d 6⊃ dδ , then d 6⊃ dδ = dϕ . Conversely, if d ⊃ dδ so that d ⊃ dδ = dϕ , then dϕ = dϕ ⊃ d ⊃ dϕ . Corollary 3.13. If δ is the generator of a dynamical system u on a full Hilbert B–module, then δ is a dδ –derivation and dδ is the generator of the C ∗ –dynamical system associated with u. In general, a d–derivation (even bounded) of a full Hilbert B–module for some derivation d of B need not be a ternary derivation, not even if d is a bounded ∗–derivation. Example 3.14. The so-called inner generalized derivations of a Hilbert B–module E are the mappings that can be written in the form δ(x) = αx − xβ a

form some α ∈ B (E) and β ∈ B. From δ(x)hy, zi+xhδ(y), zi+xhy, δ(z)i = (αx−xβ)hy, zi+xhαy−yβ, zi+xhy, αz−zβi = αxhy, zi − xβhy, zi + xhαy, zi − xhyβ, zi + xhy, αzi − xhy, zβi = δ(xhy, zi) − xhy(β + β ∗ ), zi + xhy, (α + α∗ )zi we see that δ is a ternary derivation, if and only if (β +β ∗ )hy, zi = hy, (α+α∗ )zi for all y, z ∈ E. Inserting yb for y and computing hyb, (α + α∗ )zi = b∗ hy, (α + α∗ )zi, one may check that β + β ∗ must be in the center of B. Further, the element α + α∗ ∈ Ba (E) is given simply as multiplication from the right with the central element β + β ∗ . Therefore, δ is a ternary derivation, if and only if the real parts of


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α and β may be removed without changing δ, or, in other words, if δ(x) = αx − xβ for skew-adjoint elements α and β. Notice, further, that δ is the generator of the uniformly continuous one-parameter group ut (x) = etα xe−tβ on E. It follows that this group is a dynamical system, if and only the groups etα and e−tβ are unitary. So, even if β is skewadjoint (so that d is a ∗–derivation and the generator of a C ∗ –dynamical system) δ does not generate a dynamical system, unless also α is skew-adjoint. On the other hand, if, in this case, α is not skew-adjoint, then ut is not a C0 –group. We will see in a moment that the last statement of the preceding example is typical in the sense that, if a C0 –group u consists of ϕt –linear maps ut , then u is a dynamical system. But, we think that the following preparatory result inspired very much by Lance [5, Theorem 3.5] is worth to be stated separately. Lemma 3.15. Let E be a Hilbert B–module, let F be a Hilbert C–module and suppose u : E → F is a Banach space isometry onto a C–submodule of F . If u is ϕ–linear for some homomorphism ϕ : B → C such that ϕ(B) ⊃ hu(F ), u(F )i, then u is a ϕ–isometry. Proof. For C = B, ϕ = idB and surjective u the statement is exactly [5, Theorem 3.5]. We shall prove the statement exactly along the lines of the proof of [5, Theorem 3.5] by appealing to [5, Lemma 3.4] which states b1 ≥ 0, b2 ≥ 0, and kb1 bk = kb2 bk ∀ b ∈ B

=⇒

b 1 = b2 .

First, we compute kuxk kϕ(b)k ≥ k(ux)ϕ(b)k = ku(xb)k = kxbk . If 0 6= b ∈ BE then there exists x ∈ E such that xb 6= 0. By Lemma 3.8, it follows that kϕ(b)k = kbk for all b ∈ BE . Next, for all b ∈ B and for all x ∈ E we have kϕ(b∗ )hux, uxiϕ(b)k = khu(xb), u(xb)ik = ku(xb)k

2

= kxbk

2

= kb∗ hx, xibk = kϕ(b∗ )ϕ(hx, xi)ϕ(b)k , ∗ where the last equality follows

from

b hx, xib ∈ BE and the first step. In other

p

p

words, we have hux, uxic = ϕ(hx, xi)c for all elements c ∈ ϕ(B). Since p by assumption hux, uxi ∈ ϕ(B) so that also hux, uxi ∈ ϕ(B), it follows by p p [5, Lemma 3.4] that hux, uxi = ϕ(hx, xi), hence, hux, uxi = ϕ(hx, xi) and, finally, by polarization hux, uyi = ϕ(hx, yi) for all x, y ∈ E. In other words, u is a ϕ–isometry.

Corollary 3.16. Every ϕ–linear, isometric Banach space isomorphism between full Hilbert modules with surjective ϕ is necessarily a ϕ–unitary. Remark 3.17. We do not know, whether the (necessary) condition ϕ(B) ⊃ hu(F ), u(F )i in Lemma 3.15 (and the corresponding condition ϕ be surjective of Corollary 3.16) does not, possibly, follow from the remaining hypothesis. Theorem 3.18. Suppose that d is a ∗–derivation that is the generator of a C0 –group ϕ on the C ∗ –algebra B, and suppose that δ is a d–derivation that is

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the generator of a C0 –group u on the full Hilbert B–module E. Then u is a dynamical system on E and ϕ is the C ∗ –dynamical system associated with u. Of course, δ is a ternary derivation and a d–derivation and dom(d δ ) is a core for d. Proof. For all x ∈ A(δ) and all b ∈ A(d) as in the proof of Theorem 3.1 one shows that also xb ∈ A(δ) and that ut (xb) = ut (x)ϕt (b). In exactly the same way one shows that ϕt (of course, a ∗–map) is multiplicative. In other words, ϕt is an automorphism of B and ut is a surjective and right ϕt –linear Banach space isometry. By Corollary 3.16, ut is a ϕt –unitary. In other words, u is a dynamical system and ϕ is the C ∗ –dynamical system associated with it. For the sake of clarity we summarize the criteria provided by Theorem 3.1, Corollary 3.13, and Theorem 3.18. Without the obvious proof, we add a fourth criterion based on the observation (as explained in Remark 2.6) that a dynamical system on E extends to a C ∗ –dynamical system on the linking algebra. Theorem 3.19. Let δ be the generator of a C0 –group u on a full Hilbert B–module. Then the following statements are equivalent: (1) u is a dynamical system. (2) δ is a ternary derivation. (3) There exists a ∗–derivation d that is the generator of a C0 –group on B (necessarily a C ∗ –dynamical system) such that δ is a d–derivation. (4) δ extends to the generator of a C ∗ –dynamical system on the linking algebra „ « d δ∗ of the form ∆ = δ D with δ(x∗ )∗ := δ(x)∗ and d and D being generators of C ∗ –dynamical systems on B and K(E), respectively. Remark 3.20. In all criteria where we make explicit reference to a derivation d of the corner B, we assume that both δ and d are generators of C0 –groups. We leave open the very interesting question whether the algebraic conditions alone might already be sufficient to conclude from one, δ or d, being a generator, that also the other is a generator. 4. An outline of possible applications Hilbert modules are a hybrid in between Hilbert spaces and C ∗ –algebras. Formally, the axioms of a Hilbert module over a C ∗ –algebra B generalize the axioms of a Hilbert space in that the C–valued inner product of Hilbert spaces is replaced by an inner product that takes values in B. But, also every C ∗ –algebra is a Hilbert module over itself. We have also seen that every Hilbert B–module E is a sub« „ B E∗ ∗ set of a C –algebra, the linking algebra E Ba (E) . Dynamical systems on Hilbert modules and their generators have consequences under all three aspects. While the notion of Hilbert module, formally, is a generalization of that of Hilbert space, the notion of generalized automorphism is a generalization of that of automorphism of a C ∗ –algebra. The same is true for generalized derivations. (For Hilbert spaces we recover the known notions of unitary and Stone generator of a unitary one-parameter group, respectively.) An automorphism of a C ∗ –algebra


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and a derivation of a C ∗ –algebra are a generalized automorphism and a generalized derivation, respectively, of that C ∗ –algebra when the C ∗ –algebra is considered as a Hilbert module over itself. It is noteworthy that every ϕ–unitary of the Hilbert B–module B has the form x 7→ uϕ(x), where u is a unitary in B; see [8, Corollary 1.13]. Similarly, if δ is a d–derivation of B, then δ − d defines a right linear map dom(δ) ∩ dom(d) → B. If δ is defined everywhere, then the results in [1, Section 3.1] assert that δ and d and, therefore, δ − d are bounded. We start by thinking of Hilbert modules as a generalization of Hilbert spaces. This means, in particular, we think of a Hilbert module as a space on which an operator algebra can act. The situation is particularly well under control if this algebra acting on E is the algebra of all adjointable operators Ba (E) itself. An automorphism u of the representation space E gives rise to an automorphism a 7→ uau−1 of the algebra of operators on this representation space. In the case of a Hilbert space H, the automorphisms of H are the unitaries u ∈ B(H), the corresponding automorphisms of B(H) are the inner automorphisms. It is wellknown that B(H) does not have other automorphisms than inner. Of course, also Ba (E) admits inner automorphisms, but there are more. Every ϕ–unitary u gives rise to an automorphism a 7→ uau−1 of Ba (E). This automorphisms is inner, if and only if ϕ is quasi inner ; Skeide [8, Corollary 2.3]. But the discussion following [8, Corollary 2.3] also shows that there are more (strict) automorphisms of B a (E) than the automorphisms obtained by conjugation with generalized unitaries. Summarizing, the algebra Ba (E) of all adjointable operators on a Hilbert modules is more general than B(H), but it preserves many of its features. The generalized unitaries on E are a class of operators on the Banach space E more general than unitaries, but sharing many properties of the unitaries. The class of automorphisms of Ba (E) obtained by conjugation with generalized unitaries is wider than the class of inner automorphisms but shares much of the simplicity of the latter. It seems, therefore, appropriate to try to do the whole program of finding the quantum evolution of an interacting system as a cocycle perturbations of the system without interaction for groups or cocycles of generalized unitaries. The most obvious approaches are the following two: Firstly, on can replace the unperturbed evolution of the system, usually, implement by a unitary evolution (the second quantized time shift on the Fock space, for instance) by evolutions that are implemented by conjugation with generalized unitaries, that is, by conjugation with a dynamical system. These may, then, be perturbed in the usual way by a unitary cocycle obtained via a (still to be constructed!) quantum stochastic calculus. We shall discuss in a minute a possible interpretation of this generalization of the unperturbed setting in term of an interaction picture for algebras B different from B(G). Secondly, we may try to perturb the usual dynamics by cocycles of generalized unitaries. For this we refer to the discussion in [8, Remark 3.8] and the considerations about cocycles in the end of [8, Section 1]. For the first proposal a good knowledge the dynamical systems on a Hilbert module is indispensable. For the second proposal it is necessary to find candidates for the stochastic generators of generalized unitary cocycles. We may not hope to understand these stochastic generators without understanding first the generators

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of generalized unitary cocycles with respect to the identity, in other words, the generators of dynamical systems. These notes provide the necessary theory. The second possibility how to think of generalized unitaries is that to think of an automorphism of E generalizing the notion of automorphisms of a C ∗ –algebra. The interpretation in terms of the linking algebra is closest: By Remark 2.6 (based on [8, Observation 1.4]) we know that the generalized unitaries are exactly those maps on E that extend in a “block-wise way” to an automorphism of the linking algebra, while the ternary automorphisms are exactly those maps on E that extend in a “block-wise way” to an automorphism of the reduced linking algebra (B replaced with BE ). Theorem 2.1 tells us that for full Hilbert modules the two sorts of maps are the same, while in the general case the class of ternary automorphisms is wider. The extension of u to the corner Ba (E) of the extended linking algebra is nothing but conjugation with u, a 7→ uau−1 . It is not difficult to check that the automorphism of BE we constructed in the proof of Theorem 2.1 is nothing but conjugation with the ternary automorphism x∗ 7→ (ux)∗ of E ∗ , when restricted to BE = K(E ∗ ) ⊂ Ba (E ∗ ). (It is a warmly recommended exercise to check all these assertions. The reader might ask, given a ternary automorphisms of E, why we did not immediately define the multiplicative map a 7→ uau−1 . It will help to appreciate better the proof of Theorem 2.1, if the reader tries to find out why it is not possible to prove that this defines an automorphism of Ba (E) without doing something like we did in the proof of Theorem 2.1.) Let ϕ by an automorphism of B. We see that finding a ϕ–unitary u means, first, to lift ϕ to a “generalized automorphism” u of E and, further, to an automorphism a 7→ uau−1 of Ba (E). If we describe typical applications of quantum stochastic calculus in terms of Hilbert modules, then E is the GNS-correspondence of a conditional expectation from A ⊂ Ba (E) onto B ⊂ A. That is, E is, acually, a B–bimodule (with left action B → A → Ba (E)) and there is a cyclic vector ξ ∈ E giving back the conditional expectation as a 7→ hξ, aξi. By the conditional expectation property it follows that bξ = ξb. Example 4.1. Let B = B(G) (G a Hilbert space) and E = B G, G⊗Γ(L2 (R, K)) (K some Hilbert space) the symmetric Fock module (with inner product hx, yi = x∗ y). From the left B acts directly on the factor G of G ⊗ Γ(L2 (R, K)). Then Ba (E) = B G ⊗ Γ(L2 (R, K)) and ξ : g 7→ g ⊗ Ω generates the usual vacuum conditional expectation hξ, •ξi : B G ⊗ Γ(L2 (R, K)) → B(G). If we find a ϕ–unitary u on E, then b 7→ hξ, ubu−1 ξi need no longer leave invariant the identity.

Indeed, if u is left ϕ–linear , that is, v(by) = ϕ(b)(vy), then hξ, ubu−1 ξi = ξ, uu−1 (ϕ(b)ξ) = hξ, ϕ(b)ξi = hξ, ξiϕ(b) = ϕ(b). Example 4.2. Let F be a correspondence over B (that is, a Hilbert L∞B–modules with a nondegenerate left action of B). Denote by E := F(F ) := n=0 F n the full Fock module over F , where F 0 = B and ξ = 1 ∈ B = F 0 . Let v be a left ϕ–linear ϕ–unitary on F . It is not difficult to check that the second quantization of v F 0 3 b 7−→ ϕ(b)

F n 3 yn . . . y1 7−→ vyn . . . vy1


207

defines a left ϕ–linear ϕ–unitary u on E. Of course, every such v corresponds to an even automorphism (that is, an automorphism that sends creators to creators) of the Pimsner-Toeplitz and the Cuntz-Pimsner algebras acting on F(F ). That the vacuum expectation in the usual setting is invariant under the free evolution is nothing else but the statement that we are in interaction picture. A left ϕ–linear ϕ–unitary changes this behavior. A dynamical system allows to switch from a noninteracting dynamics leaving the subalgebra B invariant, to a dynamics that restricts to C ∗ –dynamical system on B. Of course, the can be seen also in the other direction: Given a C ∗ –dynamical system on A ⊂ Ba (E) that restricts to a C ∗ –dynamical system on B ⊂ A, then finding a dynamical system u on E the the inverse C ∗ –dynamical system ϕ−1 such that every ut is also ϕ−1 t –left linear and fulfilling some cocycle condition with respect to the original dynamics on A, we may switch to a dynamics leaving B invariant by conjugation with u. We believe that this is the correct way to think of the interacting picture that generalizes the usual setting. The version of calculi we mentioned in the first part of this section should, therefore, furnish a calculus that works also if we are not in the interacting picture. Acknowledgment. The results in Section 2 were included as a part of the first authors PhD-thesis [1]. Most results in these notes have been obtained during a six months visit of the first author at the Dipartimento S.E.G.e S. financed by a grant from Ferdowsi University, Mashhad. Both authors acknowledge the support by research funds of the Italian MIUR and University of Molise. References [1] Abbaspour, G.: Dynamical systems on full Hilbert C ∗ –modules. PhD thesis, Mashhad, 2006. [2] Abbaspour, G., Moslehian, M.S., and Niknam, A.: Dynamical systems on Hilbert C ∗ – modules; Bull. Iranian Math. Soc., 31 (2005) 25–35, (arXiv: math.OA/0503615). [3] Bracic, J., and Moslehian, M.S.: On automatic continuity of 3–homomorphisms on Banach algebras; 2005, to appear in Bull. Malays. Math. Sci. Soc. [4] Bratteli, O., and Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics I (2nd ed.). Texts and Monographs in Physics. Springer, 1987. [5] Lance., E.C.: Hilbert C ∗ –modules. Cambridge University Press, 1995. [6] Muhly, P.S., Skeide, M., and Solel, B.: Representations of Ba (E); Infin. Dimens. Anal. Quantum Probab. Relat. Top., 9 (2006) 47–66, (arXiv: math.OA/0410607). [7] Skeide, M.: Generalized matrix C ∗ –algebras and representations of Hilbert modules; Mathematical Proceedings of the Royal Irish Academy, 100A (2000) 11–38, (Cottbus, Reihe Mathematik 1997/M-13). [8] Skeide, M.: Generalized unitaries and the Picard group; Proc. Ind. Ac. Sc. (Math Sc.), 116 (2006) 429–442, (arXiv: math.OA/0511661). Gholamreza Abbaspour Tabadkan: School of Mathematical Science, Damghan University of Basic Sciences, 36715-364, Damghan, Iran E-mail address: [email protected] Michael Skeide: Dipartimento S.E.G.e S., Universit¨ı> 12 degli Studi del Molise, Via de Sanctis, 86100 Campobasso, Italy E-mail address: [email protected] URL: http://www.math.tu-cottbus.de/INSTITUT/lswas/ skeide.html


SPACE REGULARITY OF STOCHASTIC HEAT EQUATIONS DRIVEN BY IRREGULAR GAUSSIAN PROCESSES OANA MOCIOALCA AND FREDERI VIENS Abstract. We study linear stochastic evolution equations driven by various infinite-dimensional Gaussian processes, some of which are more irregular in time than fractional Brownian motion (fBm) with any Hurst parameter H, while others are comparable to fBm with H < 21 . Sharp necessary and sufficient conditions for the existence and uniqueness of solutions are presented. Specializing to stochastic heat equations on compact manifolds, especially on the unit circle, sharp Gaussian regularity results are used to determine sufficient conditions for a given fixed function to be an almost-sure modulus of continuity for the solution in space; these sufficient conditions are also proved necessary in highly irregular cases, and are nearly necessary (logarithmic corrections are given) in other cases, including the H¨ older scale.

1. Introduction This article deals with precise existence results for stochastic PDEs driven by arbitrary Gaussian processes, and specializes to stochastic heat equations for sharp spatial regularity results. Since the pioneering work of stochastic analysts in the 1970’s and 1980’s (see for instance John Walsh’s Saint-Flour lecture notes [24] or DaPrato and Zabczyk’s textbook [6]), probabilists have investigated the question of how to define the weakest conditions sufficient to guarantee existence and/or regularity of a stochastic PDE’s solution. In order to express results that are as sharp as possible, the choice was made by many – including our past and present work – to study the simplest possible problems with some appeal for applications, hence the use of the stochastic heat equation driven by additive noise. The framework of It¯ o stochastic calculus was deemed most appropriate, implying the study of equations of the form Z t u (t, x) = u (0, x) + ∆x u (s, x) ds + W (t, x) (1.1) 0

for all t ≥ 0, and all x in some (e.g. Euclidean) space E, where W is some random field on R+ × E. For many years, attention was directed towards the case where W is Brownian motion in its parameter t, and it had been known since Walsh that W need not be a bonafide function for u to exist, as indeed it may be white noise in space while still allowing a solution in an analytically weak form (evolution form 2000 Mathematics Subject Classification. Primary 60H15; Secondary 60G15, 60G17. Key words and phrases. Stochastic PDE; Stochastic heat equation; Gaussian processes; fractional Brownian motion; modulus of continuity; path regularity; discontinuous processes. 209

210

OANA MOCIOALCA AND FREDERI VIENS

of DaPrato and Zabczyk), namely for E = R, with p (t, x) = (2πt)−1/2 e−x Z Z t Z p (t − s, x, y) W (ds, dy) . p (t, x − y) u (0, y) dy + u (t, x) = E

E

2

/(2t)

,

0

In the 1990’s the question of precisely how the spatial regularity of W (or lack thereof) effects the solution’s was posed. For instance, it was found that the socalled space-time white noise of the example above yields a solution with as much regularity in x as Brownian motion, so that more irregular noises should still imply existence, while qualitatively different results should be expected in higher space dimensions. Sufficient conditions were established for various additive and non-linear multiplicative stochastic PDEs: [5], [16], [17], [18], [22]. Some of these papers also covered the issue of spatial H¨ older continuity of the solution, with [22] being the first one to supply necessary and sufficient conditions for this property, and its follow-up work [23] providing an indication that extensions to non-H¨ older regularity may be possible As the case of Brownian-based noise for stochastic PDEs was now better understood, the year 2000 saw the emergence of several works focusing on the effect of fractional-Brownian-based models. These are Gaussian noise terms W (t, x) = B H (t, x) whose behavior in time has non-independent increments and a scaling property in the power H ∈ (0, 1), not simply the Brownian case H = 1/2; these are simply infinite-dimensional analogues of scalar fractional Brownian motion (fBm) introduced e.g. in [13]. The difficulty of stochastic integration with respect to such behavior in time (see [7], [1], [2]), made it so that not much progress was possible in the case of nonlinear equations: [8], [10], [11], [14]. For linear equations, however, following the impetus in [22] and [23], necessary and sufficient conditions for existence were established in a wide abstract setting in [20], and, using the stochastic heat equation on the circle, for any scale of regularity, H¨ older or not, in [21]. Still, the issue of changing the time regularity of the driving noise beyond the fractional Brownian scale has never been addressed. Only recently, in [3] and [4], has the technique for stochastic calculus for highly irregular fBm (low parameter H) been perfected. Stochastic calculus with respect to arbitrarily irregular Gaussian processes was performed in [15]. With the exception of a less than efficient treatment in [15], this article presents the first work in which the time behavior of a SPDE is neither a semimartingale nor of fractional Brownian type. The results proposed herein are systematic and quite sharp. They are formulated in a way (e.g. trace conditions in abstract Hilbert spaces: Theorem 3.3) which is consistent with the older results on semimartingale in the 1990’s, and with the new wave of fractional-Brownian-based treatments since 2000. Using infinite dimensional analogues of the processes defined in [15], we follow the framework in the work [20] to find necessary and sufficient conditions for existence of solutions to infinite-dimensional stochastic evolution equations driven by these arbitrary Gaussian fields. Our new calculation technique appears to be superior to that employed originally in [20], because we establish our existence and uniqueness theorem without needing to assume the existence of a spectral gap and

SPACE REGULARITY OF STOCHASTIC HEAT EQUATION

211

finite-dimensional kernel, which was used in [20]. Thus our corresponding work appears as an improvement over, as well as a generalization to all time scales of, the results in [20]. We apply the techniques in [21] for sharp Gaussian regularity in order to find necessary and sufficient spatial regularity conditions for the stochastic heat equation on the circle driven by arbitrary Gaussian fields. These very sharp results are owed to the use of Gaussian fields with the corresponding regularity theory introduced by Dudley, Fernique, and Talagrand, which were carefully exploited in our context in [21]. Our article is structured as follows. Preliminaries, and the definition of infinite dimensional Gaussian processes (fields) with arbitrary regularity is given in Section 2. Section 3 gives our existence and uniqueness results. Section 4 gives conditions for specific spatial regularity. While the existence result of Theorem 3.3 is stated in an abstract Hilbert-space setting, we summarized it here, together with our regularity results, using the stochastic heat equation on the unit circle S 1 , for illustrative purposes. Hypothesis: Let B γ (t, x) be a centered Gaussian random field defined for all (t, x) h∈ R+ × S 1 . Assume Biγ is homogeneous in x, with canonical metric E (B γ (1, x) − B γ (1, y))

2

= δ 2 (x − y). Let qn be the nth Fourier

coefficient of δ 2 . Note that if qn is not summable, the above definition is only formally a function, and B γ must be understood as being a Schwartzdistribution-valued Gaussian process. Assume B γ ’s behavior in time is bounded above as follows: h i 2 E (B γ (t, 0) − B γ (s, 0)) ≤ γ 2 (|t − s|) ,

where γ 2 is increasing and concave on R+ , and differentiable except at 0, with γ (0) = 0. The necessary conditions below are valid specifically if Rtp B γ (t, 0) can be written as 0 dγ 2 /dt (t − s) dB (s) where B is a standard Brownian motion. The sufficient conditions do not require this form. Conclusion for existence: [Theorem 3.3]. The stochastic heat equation (1.1) with W = B γ has a unique evolution solution in the sense of DaPrato and Zabczyk, in L2 Ω × R+ × S 1 , if and only if ∞ X

n=1

qn γ 2 n−2 < ∞.

Conclusion for regularity: [Theorem 4.2]. Let f be an increasing continuous function on a neighborhood of 0 with f (0) = 0, differentiable except at 0. Let Z r

f 0 (s) (log (1/s))

δf (r) =

−1/2

ds

0

The aforementioned solution admits f , up to a non-random constant, as a uniform modulus of continuity almost surely, if ∞ X 1 −2 2 qn γ n h δf2 0. Otherwise, the “only if” part holds with δf n−1 replaced by δf n−1 log n.

The sufficient conditions for existence and regularity hold for processes defined on R+ × R as well, if one simply replaces series by integrals above. Similar sufficient condition results also holds in higher dimensions (Rd and other compact or non-compact manifolds). We leave exact statements and proofs of these facts out of this article. 2. Preliminaries 2.1. Irregular Gaussian processes. In the remainder of the article, the symbol denotes commensurability between two functions: f and g are commensurable if there exist positive constants c, C such that cg (x) ≤ f (x) ≤ Cg (x) for all values of a common variable x. A continuous centered Gaussian process X on R+ that starts from the origin at time 0 has a distribution entirely determined by its increments’ variance structure, i.e. the canonical metric h i 2 δ 2 (s, t) = E (X (t) − X (s)) .

The case of Brownian motion X = W is δ 2 (s, t) = |t − s|, for fBm X = B H we 2H have δ 2 (s, t) = |t − s| . It is well-known that, beyond the scaling property of fBm by which B H (ct) = cH t in distribution, fBm admits the function f (r) = r log1/2 (1/r) up to a constant as a uniform modulus of continuity almost surely. The so-called Volterra representation of fBm from standard Brownian motion has the form Z t H B (t) = K (t, s) dW (s) , (2.1) 0

H−1/2

where the kernel K has the property that for s away from 0, K (t, s) |t − s| . We consider a class of Gaussian processes with arbitrary correlation between increments, by assuming that a Voterra-type representation holds, where the kernel K can be chosen to be commensurate with some given function that implies a certain type of almost-sure modulus of continuity for B H . To simplify the presentation, and as the only simple means we have found to ensure that our sufficient conditions are also necessary, we assume that K depends only on the difference t − s. All our sufficient conditions proved below hold for any other K bounded above by a given K (t − s); the proof of this fact is left to the reader. The fact that 1/2 is a motivation for the definition that follows. Also note rH−1/2 = d r2H /dr that any Gaussian process starting from 0 that is adapted to a Brownian filtration must be of the Volterra form (2.1) for an appropriate function K. Let W be a standard Brownian motion on R+ with respect to the probability space (Ω, F, P) and the filtration {Ft }t≥0 . Assume γ 2 is of class C 2 everywhere in R+ except at 0 and that dγ 2 /dr is non-increasing. In [15] it is proved that the


213

centered Gaussian process B γ (t) :=

Z

t 0

ε (t − s) dW (s) ,

(2.2)

!1/2

(2.3)

where ε (r) :=

d γ2 dr

,

satisfies the following conditions with respect to {Ft }t≥0 : for any t ≥ 0

(i) δ (s, t) γ (|t − s|), where δ is the canonical metric of B γ on (R+ ) (ii) B (0) = 0. (iii) B is adapted to a {Ft }t≥0 .

2

The choice for B γ in (2.2) above is the simplest choice satisfying conditions (i), (ii), and (iii) in terms of applications to stochastic calculus. Again, since Rt any process written as 0 K (t, s) dW (s) with K (t, s) ≤ γ (|t − s|) will satisfy the sufficient condition statements in our theorems below, our work actually covers a very wide class of Gaussian processes, and in particular reaches along the entire regularity scale of Gaussian processes. The condition that dγ 2 /dr is non-increasing implies that B γ is more irregular than standard Brownian motion, such as fBm with H < 1/2, or specifically ε (r) = rH−1/2 , which yields the so-called Liouville process, whose regularity and scaling properties are identical to those of fBm, but which has slightly inhomogeneous increments. The case where B γ is more regular than Brownian motion, such as fBm with H > 1/2, uses a slightly different, and considerably easier, calculation. We omit this case. In all cases, condition (i) above implies that up to a constant, f (r) = γ (r) log1/2 (1/r) is a uniform modulus of continuity for B γ almost surely, when lim0 f = 0. When this limit is not 0, one can prove using Gaussian supremum estimation that B γ is almost-surely discontinuous everywhere. Nevertheless, all that we claim below still holds in this extremely irregular case. 2.2. The Wiener integral with respect to B γ . Let (B γ (t))t∈[0,T ] be the centered Gaussian process defined by its Wiener integral Volterra-type representation as in (2.2) and let f be a deterministic measurable function on R+ . We define the operator K ∗ = Kγ∗ acting on f by " # Z T

Kγ∗ f (r) := f (r) ε (T − r) +

r

(f (s) − f (r)) ε0 (s − r) ds ,

if it exists. If Kγ∗ f (·) is in L2 ([0, T ], dr) then we say that f belongs to the space L2γ ([0, T ]), and we denote 2 Z T Z t

2 0 f (r) ε (T − r) + dr. kf k2γ = Kγ∗ f L2 ([0,T ]) = (f (s) − f (r)) ε (s − r) ds 0

L2γ

r

This is the so-called canonical Hilbert space of B γ on [0, T ]. We will also denote it by H. For any f in H we define the stochastic integral of f with respect

214


to B γ on [0, T ] as the Gaussian random variable given by Z T Z T f (t) dB γ (t) = Kγ∗ f (r) dW (r) . 0

0

Remark 2.1. One easily sees that if g is a function in L2γ then the function f : r 7→ g (r) 1[0,t] (r) is in L2γ and Z t K ∗ f (r) = 1[0,t] (r) ε (t − r) g (r) + [g (s) − g (r)] ε0 (s − r) ds , r

∗

i.e. K f depends on t, not on T .

2.3. Infinite dimensional irregular Gaussian processes. We define an infinite dimensional Gaussian process using the classical approach of [6] for cylindrical Brownian motion (see section 4.3.1 therein). Let U be a real separable Hilbert space. For any fixed complete orthonormal basis (en )n in U and any fixed sequence P of positive numbers (λn )n , even if n≥0 λn = ∞, we define B γ (t) =

∞ p X

λn en Bnγ (t),

(2.4)

n=0

where Bnγ are an IID sequence of Gaussian processes with the same distribution as the B γ in the previous section. This slight abuse of notation will not lead to confusion, since henceforth B γ denotes an infinite-dimensional process. Observe that for any fixed t the above series converges in L2 (Ω × U ) if and only if P γ n≥0 λn < ∞. In the other case, B (t) will be a well defined Gaussian process with values in a larger Hilbert space U1 , where the embedding U ⊂ U1 is continuous, Hilbert-Schmidt. For instance, if U = L2 ([0, 1]), we can use the space of Schwartz tempered distributions for U1 . To define the Wiener integral with respect to the above infinite dimensional Gaussian process B γ we consider another real separable Hilbert space V and (φs )s∈T a deterministic function with values in L2 (U ; V ), the space of HilbertSchmidt linear operators from U to V . The stochastic integral of φ with respect to B γ is defined by Z t ∞ Z t ∞ Z t X X (K ∗ φen )s dBn (s), φs en dBnγ (s) = φs dB γ (s) = 0

n=0

0

n=0

0

where Bn is the standard Brownian motion used to represent B γ in the Volterratype representation (2.1). The above sum is finite almost surely, and is indeed a Gaussian random element of V , if and only if it has a finite variance in V , i.e. X kK ∗ (φen )k2L2 ([0,T ],V ) < ∞. n

3. Linear stochastic equations. Existence of solutions Consider the setting from the previous section where B γ a cylindrical Gaussian process (a process defined as above for which λn = 1), and φ a linear operator in L(U, V ) that is not necessarily Hilbert-Schmidt.


215

We will study the existence of solutions for the equation dX(t) = AX(t)dt + φdB γ (t),

(3.1)

with boundary condition X(0) = x ∈ V , where A : Dom(A) ∈ V → V is the infinitesimal generator of the strongly continuous semigroup (etA )t∈T , a self adjoint operator on V . Rt Remark 3.1. This equation is defined if the integral 0 φdB γ is defined as a V valued random variable. Since 2 Z t 2 X Z t γ γ E φen dBn (s) E φdB (s) = 0

V

n

0

V

2 X Z t = E dBnγ (s) |φen |2V = γ 2 (t)kφk2HS , n

0

this occurs if and only if φ ∈ L2 (U, V ).

Nevertheless, it is standard practice in stochastic analysis, useful in applications, to consider a weak form of the above differential equation, well-defined even for many non-Hilbert-Schmidt integrands φ, depending on the regularity of the operator A. Definition 3.2. The mild, or evolution, form of the stochastic differential equation (3.1), with starting point X (0) = x ∈ V , is given as follows: for all t ∈ [0, T ], we have the equality in V almost surely: Z t tA e(t−s)A φdB γ (s). X(t) = e x + 0

Since this is an explicit formula for X, we call it the mild, or evolution, solution to (3.1). Reed and Simon in [19] presented the existence of an uniquely defined projection measure dPλ on the real line, such that for every φ ∈ V , dhφ, Pλ φi is a Borel measure on R, and for every φ ∈ Dom(A) Z hφ, Aφi = λdhφ, Pλ φi. R

The next theorem is a generalization of Theorem 40 in [15].

Theorem 3.3. Let ε and γ be defined as in Section 2 (see definitions (2.2) and (2.3)). Assume |ε0 (x)| 1 ≤ (3.2) ε(x) 2x and 3 |ε0 (x)| x− 2 f (x) (3.3) γ hold, where f is an increasing differentiable function, and let B be a cylindrical Gaussian process on the Hilbert space U . Assume A : Dom(A) ∈ V → V is a negative, self-adjoint operator on the Hilbert space V . Note that we do not need to assume A is negative-definite.

216


Then for any fixed φ ∈ L(U, V ) there is a unique mild solution X(t) ∈ L2 (Ω, V ) for the equation dX(t) = AX(t)dt + φdB γ (t), t ∈ [0, T ]; X(0) = x ∈ V

if and only if φ∗ GH (−A)φ is a trace-class operator, where 1 2 . GH (λ) = γ max(λ, 1)

R ∞Here for any integrable function F on R+ , F (−A) is defined by F (−A) x = 0 F (λ) P−λ xdλ for all x ∈ V .

Remark 3.4. Our conditions on ε are not restrictive. Since ε2 is integrable near 0 due to definition (2.2), we can assume without loss of generality that ε2 (x) x is increasing near 0. This means 2ε (x) ε0 (x) x + ε2 (x) ≥ 0, which implies (3.2) for all x near 0. Remark 3.5. Condition (3.3) signifies how to compare the Volterra kernel with those standard power scale ones, with γ (x) = xH , so that |ε0 (x)| = cH x−3/2 xH , or logarithmic scale ones, with γ (x) = log−β (1/x), so that indeed |ε0 (x)| x−3/2 log−β (1/x), from which we see that requiring f increasing is not a restriction in any scale. Proof of the theorem. Step 1: setup and exact calculations. Consider the scalar measure µn defined as dµn (λ) = dhφen , Pλ φen iV . Denoting It = E|X(t) − etA x|2V ,

it is sufficient to estimate It optimally from above and below. By independence of the components in the definition (2.4) of the infinite-dimensional B γ , we have Z t 2 tA 2 (t−s)A γ It = E|X(t) − e x|V = E e φdB (s) 0

V

2 X Z t (t−s)A γ e φen dBn (s) . = E n 0 V

Then using the definition of Wiener integration in Section 2.3, 2 X Z T h i ∗ (t−s)A It = E K e φen 1[0,t] (s) dW (s) 0 n V Z i T h X 2 ∗ = Kγ 1[0,t] e(t−s)A φen ds n

0

V

2 Z t XZ t (t−s)A (t−r)A (t−s)A 0 = φen ε(t − s) + (e φen − e φen )ε (r − s)dr ds. e n

0

s

V


217

For ease of computations we can take t = 1. Then, by definition of esA , I1

(3.4) 2 Z 1 (1−s)A e φen ε(1 − s) + (e(1−r)A φen − e(1−s)A φen )ε0 (r − s)dr ds = s 0 V n Z Z Z 1 ∞ 1 X = e−(1−s)λ ε(1 − s) + (e−(1−r)λ − e−(1−s)λ )ε0 (r − s)dr dsdµn (λ) 1

XZ n

=

XZ n

0

0

∞

0

s

(Z

1

0

"

−λ(1−s)

=

+2e XZ ∞ n

e

Z

−2λ(1−s) 2

ε (1 − s) +

ε(1 − s)

Z

1

(e

−λ(1−r)

s

−e

−λ(1−s)

1

(e

−λ(1−r)

s

−e

−λ(1−s)

)ε (r − s)dr

0

0

2

)ε (r − s)dr ds dµn (λ)

I(λ)dµn (λ),

(3.5)

0

where

I(λ) =

Z

1 0

"

e

−2λ(1−s) 2

+2e =

Z

ε (1 − s) +

−λ(1−s)

1

e−2λ(1−s) 0

ε(1 − s) (

= = =

Z Z

1 0

(e

s

Z

e

−e

−λ(1−s)

0

)ε (r − s)dr

−λ(1−r) 1 s

−e

−λ(1−s)

ε(1 − s) +

1

s

)ε (r − s)dr ds

(eλ(r−s) − 1)ε0 (r − s)dr

(eλ(r−s) − 1)ε0 (r − s)dr

1−s

(e 0

uλ

0

− 1)ε (u)du

2 Z s (euλ − 1)ε0 (u)du ds. e−2λs ε(s) +

2

0

(e−λ(r−s) − 1)ε0 (r − s)dr ds Z

−2λ(1−s)

s

Z

e−2λ(1−s) ε(1 − s) +

0

0

1

(e

−λ(1−r)

s

1

1

Z

1

1

ε2 (1 − s) +

+2ε(1 − s) Z

Z

Z

2

2

2

ds

ds (3.6)

0

The equality before the last one is obtain by the change of variable u = r − s and the last one with the change of variable s = 1 − s. Estimation of the formula in (3.6) is non-trivial: p indeed the two terms in the square inside I (λ) are of opposite signs, since ε (r) = dγ 2 /dr is assumed to be decreasing. In steps 2 and 3 below, we assume λ ≥ 1. Step 2: upper bound when λ ≥ 1. We first establish an upper bound on I (λ). We observe that "Z Z s 2 # Z 1 1 (euλ − 1)ε0 (u)du ds I(λ) ≤ 2 e−2λs ε2 (s)ds + e−2λs 0

0

0

218


= I1 (λ, 1) + I2 (λ, 1) This is not a sharp inequality a priori, since it kills the negativity of ε0 . Nevertheless, our lower bound below shows that it is actually sharp. Step 2a: upper bound, 2nd term. We start by analyzing the second term of the inequality Z s 2 Z 1 I2 (λ, 1) = e−2λs (euλ − 1)ε0 (u)du ds 0

≤

Z

0

1 λ

e−2λs

0

+

+

Z Z

Z

s

0

(euλ − 1)ε0 (u)du

1

e

−2λs

Z

2

1 λ

1

e

−2λs

Z

2

1 λ

1 λ

(e

uλ

0 s

(e

uλ

1 λ

2

ds

0

− 1)ε (u)du 0

− 1)ε (u)du

!2

!2

ds

ds

:=I2,0 (λ) + I2,1 (λ) + I2,2 (λ), and now bounding each of these three terms. To control the first term we bound eλr−1 above by Cλr and 2−2λs by 1. The actual value of the constant C below may change from line to line, but never depends on λ. 2 Z 1/λ Z s 2 0 I2,0 (λ) ≤ C λ r |ε (r)| dr ds. 0

0

Keeping in mind the special condition (3.2) we obtain 2 Z λ1 Z s 2 ε(r)dr ds I2,0 (λ) ≤ Cλ 0

≤ Cλ = Cλ

2

Z

0

1 λ

Z

0

Z

1 λ

ε(r)dr 0

1 λ

ε(r)dr 0

!2

!2

ds

.

For the second term, using the same approximations and inequality as above, we obtain !2 Z 1 Z 1/λ I2,1 (λ) ≤ C e−2sλ λr |ε0 (r)| dr ds 1/λ

≤C

Z

0

1

e−2sλ 1/λ

λ

Z

1/λ

ε (r) dr 0

1 −2 =C e − e−2λ λ 2λ

Z

!2

ds

1/λ

ε (r) dr 0

!2


e−2 λ ≤C 2 Z

= Cλ

Z

1 λ

ε(r)dr 0

1 λ

ε(r)dr 0

!2

219

!2

.

The last term can be evaluated as in [15]. It was shown that if ε0 has the representation |ε0 (r)| r−3/2 f (r) with f differentiable and increasing, and |ε0 | decreasing, then 1 I2,2 ≤ Cf 2 ( ). λ We can rewrite this as 1 1 I2,2 ≤ Cf 2 ( ) = Cλ3 λ−3 f 2 ( ) λ λ !2 − 32 1 1 1 1 1 ≤ C 3 (ε0 )2 ( ) f( ) =C 3 λ λ λ λ λ 1 1 1 1 0 2 1 (ε ) ( ) ≤ C ε2 ( ) 2 λλ λ λ λ 2 1 1 = Cλ ε( ) λ λ !2 Z λ1 ε(r)dr , ≤ Cλ =C

0

where the last inequality was obtain by the monotonicity of ε. Putting the bounds of the three terms together we obtain !2 Z λ1 ε(r)dr . I2 (λ, 1) ≤ I2,0 (λ) + I2,1 (λ) + I2,2 (λ) ≤ Cλ 0

Step 2b: upper bound, first term. For the upper bound of the first term in the 1 evaluation of I(λ), I1 (λ, 1), we use the fact that for s ≥ λ1 we have e2λs ≤ (λs) 2, we use a scalar change of variables and the monotonicity of ε in order to obtain Z 1 Z λ1 Z 1 1 2 I1 (λ, 1) = e−2λs ε2 (s)ds ≤ ε2 (s)ds + ε (s)ds 2 1 (λs) 0 0 λ Z Z λ1 1 λ 1 2 u ε2 (s)ds + = ε ( )du λ 1 u2 λ 0 Z λ1 Z λ 1 1 1 ε2 (s)ds + ε2 ( ) ≤ du λ λ 1 u2 0 Z λ1 1 1 ε2 (s)ds + ε2 ( ) ≤ λ λ 0 !2 Z λ1 Z λ1 ≤ ε2 (s)ds + λ ε(s)ds . 0

0

220


Again using the assumption 1 |ε0 (x)| ≤ , ε (x) 2x we have Z

x 0

which implies

ε (r) dr ≥ 2

Rx 0

x

Z

|ε0 (r)| rdr = −2ε (x) x + 2

0

Z

x

ε (r) dr,

0

ε (r) dr ≤ 2ε (x) x and in particular Z λ1 1 . λ ε (r) dr ≤ 2ε λ 0

Then using the monotonicity of ε, !2 Z λ1 Z λ1 Z λ1 1 1 γ2( ) = ε2 (r)dr ≥ ε( ) ε(r)dr ≥ Cλ ε(r)dr . λ λ 0 0 0 This completes the upper bound proof since, now !2 Z λ1 1 1 1 2 1 I(λ) ≤ γ ( ) + Cλ ≤ γ 2 ( ) + Cγ 2 ( ) = Cγ 2 ( ). ε(r)dr λ λ λ λ 0 Step 3. Lower bound when λ ≥ 1. For the lower bound there is a simple strategy. It is certainly true that 1/ (2λ) ≤ 1, and from (3.6) we obtain the trivial lower bound 2 1 Z s Z 2λ (euλ − 1)ε0 (u)dr ds. e−2λs ε(s) + I(λ) ≥ 0

0

Since the two terms inside the square are of opposite sign, our strategy is to show that the second term (in absolute value) is less than a constant K times the first term, with K < 1: if we have Z s (euλ − 1)|ε0 (u)|dr ≤ Kε(s), (3.7) 0

then

ε(s) −

Z

s 0

(euλ − 1)|ε0 (u)|dr ≥ (1 − K)ε(s),

which implies, using the fact that ε decreases, 1 Z 2λ 2 I(λ) ≥ e−2λs ((1 − K)ε(s)) ds 0

= (1 − K)

Z

≥ (1 − K) e

1 2λ

e−2λs ε2 (s)ds

0

−1

≥ (1 − K) e−1

Z Z

1 2λ

ε2 (s)ds

0 1 2λ

0

ε2 (2s)ds


221

1 λ

1 ε2 (s)ds = (1 − K) e−1 γ 2 ( ), λ 0 which is all that is needed for the proof of the lower bound. 2 To establish this we use again the special condition (3.2). Since ε is integrable 2 −1 at the origin, it holds that ε (r) = o r and thus we can also assume without loss of generality, similarly to condition (3.3) (see Remark 3.5), that there exists 1 an increasing function g with with ε(r) = r − 2 g(r). Thus we get Z s Z λ s euλ − 1 (euλ − 1)|ε0 (u)|dr ≤ ε(u)du 2 0 uλ 0 ≥ (1 − K) e

−1

Z

√

x

Since the function e x−1 is bounded for x ∈ [0, 12 ], by 1 below and e−1 above, we 2 get √ Z Z s e−1 s uλ 0 ε(u)du (e − 1)|ε (u)|dr ≤ λ √ 2 2 0 0 √ √ Z s √ e−1 e−1 − 21 u g(u)du ≤ λ √ g(s)2 s =λ √ 2 2 0 2 2 √ e−1 = √ λsε(s) 2 √ e−1 ≤ √ ε(s). 2 2 √

√ < 1 this completes the proof for the lower bound when λ ≥ 1. Since 2e−1 2 Step 4: conclusion when λ ≥ 1. From the results of Steps 2 and 3, we have proved that for any λ ≥ 1, 1 2 1 2 . I (λ) γ ( ) = γ λ max (λ, 1)

Step 5: case λ ∈ [0, 1]. A precise estimate of I (λ) is more difficult in this case, but we do not need to have a precise result. Indeed, we only need to show that for all λ ∈ [0, 1], 1 2 I (λ) γ = γ 2 (1) . max (λ, 1)

In other words, we only need to show that I (λ) is bounded above and below by positive constants, uniformly in λ ∈ [0, 1]. Using ex − 1 ≤ ex for x ∈ [0, 1], using ε0 < 0, and integrating by parts (using the fact that ε (s) = o s−1/2 ), the negative term (with the ε0 ) in formula (3.6) is bounded above as 2 Z s 2 Z s 0 uλ 2 0 e − 1 ε (u) du ≤ e uλε (u) du 0 0 Z s 2 2 2 =e λ (ε (u) − ε (s)) du Z0 s 2 2 ε2 (u) du = e2 sλ2 γ 2 (s) . ≤ e sλ 0

222


where we used Jensen’s inequality in the last step. Therefore we immediately have the upper bound Z 1 Z 1 I (λ) ≤ 2ε2 (s) ds + 2e2 sλ2 γ 2 (s) ds 0 0 ≤ 2 + 2e2 γ 2 (1) . For a lower bound, define

f (λ, s) = ε(s) +

Z

s 0

(euλ − 1)ε0 (u)du.

(3.8)

R1 From (3.6), we see that I (λ) ≥ e−2 0 f 2 (λ, s) ds. A positive lower on I (λ) uniform for all λ ∈ [0, 1] now follows from the lemma below. Step 6: final lemma, and conclusion. Since the results above, including the next lemma, establish that I (λ) γ 2 (1/ max (λ, 1)) for all λ ∈ R+ , the theorem follows; indeed we can assert XZ ∞ I1 γ 2 (1/ max (λ, 1)) dµn (λ) = tr (φ∗ GH (−A)φ) . 0

n

Lemma 3.6. With f (λ, s) as in (3.8), min

nR

1 0

o

f 2 (λ, s) ds : λ ∈ [0, 1] > 0.

Proof. First note that f is differentiable with respect to s everywhere except at 0, and that we have ∂f (λ, s) = ε0 (s) esλ < 0 ∂s so that f (λ, ·) is decreasing on (0, 1]. We have lims→0 ε (s) = +∞, and we proved in Step 5 that Z s lim euλ − 1 ε0 (u) du ≤ lim e2 sγ 2 (s) = 0. s→0

s→0

0

Therefore lims→0 f (λ, s) = +∞. Hence for each λ ∈ [0, 1], there exists a value s∗ (λ) ∈ (0, 1] such that f (λ, s) ≥ 1 for all s ≤ s∗ (λ), and define s∗ (λ) to be maximal such. Note that for those values of λ for which f (λ, s) exceeds 1 for all s ∈ [0, 1], this simply means that the corresponding s∗ (λ)’s are all equal to 1. Moreover, we calculate Z s ∂f (λ, s) = ueuλ ε0 (u) du < 0, ∂λ 0

so that s∗ is non-increasing. This means that, defining s∗∗ = s∗ (1) which is strictly positive as noted above, we have for all s ≤ s∗∗ , for all λ ∈ [0, 1], f (λ, s) ≥ 1, and we finally obtain Z 1 Z s∗∗ f 2 (λ, s) ds ≥ f 2 (λ, s) ds ≥ s∗∗ > 0, 0

finishing the proof of the lemma.

0


223

4. Functional equations and space regularity 4.1. Evolution equations on manifolds. The abstract framework of Theorem 3.3 is useful in a number of more concrete situations. We will illustrate this point by investigating the so-called additive stochastic heat equation, namely a parabolic stochastic PDE of the form Z t u (t, x) = u0 (x) + ∆x u (s, x) ds + W (t, x) (4.1) 0

for some Gaussian random field W on the cartesian product of R+ × M where M is a finite-dimensional space where ∆x can be defined; W could thus range over a wide array of infinite-dimensional versions of our B γ (t) defined by (2.2). Using again the evolution interpretation in the manner of Da Prato and Zabczyk [6], we replace (4.1) by Z t u (t, x) = Pt u0 (x) + Pt−s W (ds, ·) (x) , (4.2) 0

where (Pt )t≥0 is the semigroup of operators generated by ∆x . Indeed, a solution to (4.1) also solves (4.2), but the latter is considerably weaker, since the Wiener integral above may exist even if W is not a bonafide function in x. More specifically, for an arbitrary smooth compact Riemannian manifold M and its Laplace-Beltrami operator ∆x , let (λn , en )n∈N be the eigenvalues and eigenfunctions of −∆x which we can arrange in increasing order with λ0 = 0 and λn > 0 for all n > 0; then under the Riemannian inner product, {en }n∈N can be chosen as an orthonormal basis for a Hilbert space of functions V on M , the space of square-integrable functions with respect to the Riemannian volume element. We use for W the random field B γ defined formally by X√ qn en (x) B γ (t) , B γ (t, x) = n∈N

(Bnγ )n∈N γ

P where is a family of independent copies of our B γPin (2.2). If n qn is finite, this B (t, ·) is a V -valued Gaussian element, but if n qn is infinite, this definition is only formal, and in reality B γ (t, ·) is typically distribution-valued. Theorem 3.3 is easier to express in this framework because ∆x and the spatial covariance of B γ are both diagonalizable in the basis of V , and also ∆x has a spectral gap. Theorem 3.3 implies that Z t u (t, x) = Pt u0 (x) + Pt−s B γ (ds, ·) (x) (4.3) 0

has a solution u (t, ·) in V if and only if X 1 2 qn γ < ∞, λn

(4.4)

n∈N

and in this case the solution is a Gaussian random field on R+ × M , and is also a V -valued Gaussian stochastic process. Since limr→0 γ (r) = 0, there are obviously solutions of (4.3) corresponding to non-summable sequences qn , i.e. to fields B γ which are not bonafide L2 functions in x. This is the usual observation for additive

224


stochastic PDEs, generalized here to all scales of potential irregularity in time, and there seems to be little to gain by singling out the case of summable qn . The above development works also for non-compact manifolds, such as Rd , where all above statements can be written using integrals with respect to n2 = λ ∈ R+ instead of series over n ∈ N, but the absence of a spectral gap for the Laplacian forces one to revert to expressions involving γ 2 (1/ max(λ, 1)) instead of simply γ 2 (1/λ). We leave these details out. Returning to the compact case, if condition (4.4) is satisfied more than just barely, one should expect some regularity for the solution in (4.3). We now illustrate this phenomenon in the specific example of the circle. 4.2. Solution regularity on the unit circle. Assume now that M = S 1 , the circle parametrized by [0, 2π). Therefore, it is most convenient to represent the eigenstructure of the Laplace-Beltrami ∆x by saying that for each n ∈ N the eigenvalue λn = −n2 has two eigenfunctions en (x) = cos (nx) and fn (x) = sin (nx). Let now B γ be as above, and assume in addition that it is homogeneous in the parameter x, a with a given covariance structure Q in space, which means that it can be represented as Z t B γ (t, x) = ε (t − s) W (ds, x) , 0

where the Gaussian field W on R+ × S 1 has covariance E [W (t, x) W (s, y)] = Q (x − y) min (s, t). It is then possible to express the decomposition of B γ in the 1 2 trigonometric basis of V = L S , i.e. as a Gaussian Fourier series: B γ (t, x) =

√

q0 B0γ (t) +

∞ ∞ X X √ ¯γ √ qn Bn (t) sin (nx) + qn Bnγ (t) cos (nx) , (4.5)

n=1

n=1

¯nγ are independent families of independent copies of the B γ where and B n in (2.2), and (qn )n is a sequence of non-negative numbers. In fact, since Q is a positive definite function on S 1 , and thus a member of L1 S 1 , the values qn are easily seen to be its Fourier coefficients. Since sin (nx) and cos (nx) share the eigenvalue exp −n2 t for Pt , we can immediately rewrite (4.3), assuming without loss of generality that u0 ≡ 0, that Z t Z t ∞ X 2 2 √ √ e−(t−s)n dB0γ (s) + u (t, x) = q0 qn cos (nx) dBnγ (s) e−(t−s)n (Bnγ )n

0

+

∞ X

0

n=1

√ qn sin (nx)

n=1

Z

t

0

¯ γ (s) e−(t−s)n dB n

2

with, by Theorem 3.3, existence and uniqueness holding if and only if X qn γ 2 n−2 < ∞.

(4.6)

(4.7)

n∈N

More precisely, since the proof of Theorem 3.3 translates here as nothing more than an estimation of the variances of the Gaussian random variables in (4.6) from above and below, and (4.6) clearly shows that u (t, ·) is spatially homogeneous, we have actually proved the following.


225

Corollary 4.1. When V = L2 S 1 , with the V -valued Gaussian process B γ defined in (4.5), and u the solution (4.3) of the stochastic heat equation driven by B γ , expressed for example in (4.6), for every fixed t ∈ [0, 1], u (t, ·) is a homogeneous Gaussian field on S 1 satisfying Xp ¯n , u(t, ·) = sn (t) cos (n, ·) Gn + sin (n, ·) G n∈Z

¯ are independent sequences of IID standard normal random variwhere G and G ables, and sn (t) qn γ 2 (n−2 ).

The commensurability constants depend on t, γ and the sequence {q n }n∈N , but not on n. This corollary is all that is needed to apply the regularity results in Section 3.3 of [21]. We leave the proof of the theorem below, which is no more than bookkeeping, to the reader. It gives sufficient, and largely necessary, conditions for the solution u to have a prescribed almost-sure modulus of continuity in space. Let Y be the Gaussian random field defined on S 1 by Y (x) =

∞

X√ √ q 0 Z0 + qn γ n−2 Zn sin (nx) + Z¯n cos (nx) ,

(4.8)

n=1

where Z and Z¯ are independent sequences of IID standard normal r.v.’s. Since Y is clearly a homogeneous Gaussian field on S 1 , we can calculate its homogeneous canonical metric function δY : δY2

h

(x − y) = E (Y (x) − Y (y))

2

i

=

∞ X √

n=1

qn γ n−2 (1 − cos (n (x − y))) . (4.9)

Consider then the function f = fδY defined on a neighborhood of 0 via the rule Z ∞ Z δ(α) q 2 f (α) = fδ (α) := δ min e−x , α dx = log 1/δˇ (ε)dε, 0

0

where δˇ is the inverse function of δ. From the work of Fernique [9], which interprets the so-called Entropy upper bound of Dudley (see [12]), we know that the function f , if its limit is 0 at 0, is an almost-sure uniform modulus of continuity for Y , i.e. that |Y (x) − Y (y)| : x, y ∈ S 1 sup (4.10) f (|x − y|)

is almost-surely finite. The following theorem, established exactly like Theorem 4 in [21], shows that f is also an almost-sure uniform modulus of continuity for u (t, ·); it gives a way to construct a Y and a u (t, ·) which share a given function f as an almost-sure uniform modulus of continuity, by ensuring a convergence condition on the coefficients qn ; it even shows a converse is true in the sense that if u (t, ·) has f as an almost-sure uniform modulus of continuity, then the convergence condition should hold.

226


Theorem 4.2. Let f be an increasing continuous function on a neighborhood of 0 in R+ , continuously differentiable everywhere except at 0, with lim0+ f = 0. Let Y and u be given by (4.8) and (4.6). Let δf be given by Z α −1/2 −3/2 −1 δf (α) = f (α) (log (1/α)) − f (r) (log (1/r)) (2r) dr. 0

It is not necessary to assume that this function δf has the form in (4.9). It is, however, positive and increasing. Sufficient Condition: Assume that for any continuous, decreasing, differR1 entiable function h on [0, 1] with 0 h (x) dx < ∞, 2 ! X 1 2 −2 qn γ n h δf < ∞. (4.11) n n Then f is an almost-sure uniform modulus of continuity for both Y and u (t, ·), in the sense of (4.10). Necessary condition: sharp case: When f (r) r H for any H > 0, the converse is true. Namely, assume f is an almost-sure uniform modulus of continuity for Y or for u (t, ·); then (4.11) holds. Necessary condition: Holder case: When it is not true that f (r) r H holds for all H > 0, the converse holds up to a logarithmic correction. Namely, assume f is an almost-sure uniform modulus of continuity for Y or for u (t, ·); then (4.11) holds with δf (1/n) replaced by δf (1/n) log (n).

It should be noted that the canonical metrics of Y and u (t, ·) are, up to constants, bounded above by δf . A similar theorem which, instead of Condition (4.11), uses the condition that Y admits f as an almost-sure uniform modulus of continuity, also holds. See Theorem 3 in [21]. We finish this article with some examples of the precision allowed by the above theorem. 4.3. Examples on the unit circle. 4.3.1. Fractional Brownian scale. In the fractional Brownian scale, where γ (r) rH0 , the logarithmic correction is not visible in the H¨ older scale, because the correction needed to make the function h (r) = 1/r integrable at the origin is also 0 logarithmic, and because the ratio of f (r) = r H over the corresponding δf is again in the logarithmic scale. Therefore we can state the following necessary and sufficient condition. The solution u to (4.7) is almost surely H 0 -H¨ older-continuous in x for all H 0 < H1 if and only if, for all H 0 < H1 , X 0 qn n−4H0 +2H < ∞. n

00 √ For instance, if qn n−1/2−H , we get

H 00 > H1 − 2H0 .

(4.12)

To be more precise, including the logarithmic terms, and using a general γ, to get that u is precisely H1 -H¨ older continuous, i.e. to get f (r) = r H1 , we see


227

that δf (r) ≤ rH1 log−1/2 (1/r), so it is sufficient to choose qn such that (using h (r) = r−1 log−1 (1/r) (log log)−1 (1/r)), X ∞> qn γ 2 n−2 n2H1 · log n · log−1 n2H1 log n (log log)−1 n2H1 log n n

X n

−1 qn γ 2 n−2 n2H1 (log log) (n) .

Similarly, to obtain a u which has exactly the same regularity in space as the fBm with parameter H1 , we need f (r) = r H1 log1/2 (1/r). Thus we get δf (r) ≤ rH1 , and we only need to require that, X −1 ∞> qn γ 2 n−2 n2H1 log−1 (n) (log log) (n) . n

Because of the logarithmic correction needed to make the converse work, we can only state, for instance, that if f (r) = r H1 is an almost-sure modulus of continuity for u in space, then for any α > 1, X qn γ 2 n−2 n2H1 log−2 (n) (log log)−α (n) < ∞, n

−2

and the log n above should be replaced by log−1 n if we only know that u has the same regularity in space as fBm with parameter H1 . 4.3.2. Logarithmic regularity scale. The case of Gaussian fields whose almostsure modulus of continuity is commensurate with f (r) = log−β (1/r) for β > 0, which we like to call the logarithmic Brownian scale, coincides, according to our statements regarding Y in the above theorem, with δf (r) log−β−1/2 (1/r), and coefficients qn satisfying, up to a triply iterated logarithmic term, X −1 qn γ 2 n−2 log2β (n) (log log) (n) < ∞. n

More precisely, the above condition is sufficient for u to have f (r) = log−β (1/r) as a uniform modulus of continuity in x, but if the latter holds, then the above −α series converges if one adds a factor (log log log) (n) for any α > 1. When B γ itself is in the logarithmic Brownian scale in time, meaning γ (r) log−β0 −1/2 (1/r) for some β0 > 0, we find that log−β1 (1/r) is a uniform modulus of continuity for u in x as soon as X −1 qn log2β1 −2β0 −1 (n) (log log) (n) < ∞ n

with the condition being necessary if a factor (log log log) indeed the necessary and sufficient condition is simply that X qn an log2β1 −2β0 (n) < ∞

−α

(n) is added, and

n

for all positive sequences {an }n∈N such that n−1 an is summable. For instance, if 00 √ we assume that qn n−1/2 log−2β (n), we see that we only need to take β 00 ≥ β1 − β0 .

(4.13)

228


4.3.3. Conclusion. While the last result may seem esoteric, it actually has an important interpretation, when compared to (4.12). First we have the fact that (4.13) is more precise than (4.12) – we have an exact upper bound on β 00 , not a gap as required in (4.12). But more importantly condition (4.12) indicates that to obtain a H1 -H¨ older-continuous solution u, the H¨ older-continuity of B γ 00 γ in space (measured by H ) has to be combined with B ’s H¨ older-continuity in time (measure by H0 ), but that the latter is twice as strong as the former; this is a phenomenon familiar to those who know that for the standard stochastic heat equation with infinite-dimensional Brownian potential (here γ (r) = r −1/2 ), when the spatial regularity of B γ is such that the solution is H-H¨ older-continuous in space, then it is only H/2-H¨ older-continuous in time. The situation in the logarithmic scale is not the same. Condition (4.13) shows that the combined logarithmic continuity of B γ in space and time are to be compared with equal weights (β0 + β 00 ), i.e. without the factor 2 in time, with the solution’s logarithmic continuity. In conclusion, the common intuition saying that the stochastic heat equation’s regularity is twice as strong in space as it is in time, the factor 2 being due to the quadratic variation of Brownian motion, is misleading. We see here that, in the H¨ older scale, the effect of the potential B γ ’s time regularity is always twice as heavy as its space regularity, that this appear to be a general property of the heat equation since it has nothing do to with the presence of white-noise in time, as it holds for all γ (r) r H0 , not just H0 = 1/2. But on the other hand, the relative strengths of the potential’s time regularity is equal, not double, its space regularity, in the logarithmic regularity scale, which means that the type of noise can make a difference in how the potential’s regularity effects on the heat equation, even though one has to reach to logarithmic regularity to deviate from the familiar rule by which a potential’s time regularity effect’s the solution twice as strongly as its space regularity.

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[10] Grecksch , W. and Ahn, V. V.: A parabolic stochastic differential equation with fractional Brownian motion input; Statistics and Probability Letters 41 (1999) 337–346. [11] Kleptsyna, M. L., Kloeden, P. E., and Ahn, V.V.: Existence and uniqueness theorems for stochastic differential equations with fractal Brownian motion; (Russian) Problemy Peredachi Informatsii 34 (1998) no. 4, 51–61; translation in Problems Inform. Transmission 34 (1998), no. 4, 332–341 (1999). [12] Ledoux, M. and Talagrand, M.: Probability on Banach Spaces. Birkhauser, 1990. [13] Mandelbrot, B. B. and Van Ness, J. W.: Fractional Brownian motion, fractional noises and application; SIAM Review 10 (1968) 422–437. [14] Maslowski, B. and Nualart, D.: Evolution equations driven by a fractional Brownian motion; J. Functional Analysis 202 (2003) 277–305. [15] Mocioalca, O. and Viens, F.: Skorohod integration and stochastic calculus beyon the fractional Brownian scale; J. Functional Analysis 222 (2005) 385–434. [16] Millet, A. and Sanz-Solé, M.: A stochastic wave equation in two spatial dimensions: smoothness of the law; Ann. Probab. 27 (1999) 803–844. [17] Peszat, S. and Zabczyk, J.: Stochastic evolution equations with a spatially homogeneous Wiener process; Stochastic Process. Appl. 72 (1997) 187–204. [18] Peszat, S. and Zabczyk, J.: Nonlinear stochastic wave and heat equations; Probab. Theory Related Fields 116 (2000) 421–443. [19] Reed, M. and Simon, B.: Functional Analysis I. 2nd edition, Academic Press, 1980. [20] Tindel, S., Tudor, C. A., and Viens, F.: Stochastic evolution equations with fractional Brownian motion; Probab. Theory Related Fields 127 (2003) 186–204. [21] Tindel, S., Tudor, C. A., and Viens, F.: Sharp Gaussian regularity on the circle, and applications to the fractional stochastic heat equation; J. Functional Analysis 217 (2004) 280–313. [22] Tindel, S. and Viens, F.: On space-time regularity for the stochastic heat equation on Lie groups; J. Functional Analysis 169 (1999) 559–603. [23] Tindel, S. and Viens, F.: Regularity conditions for parabolic SPDEs on Lie groups; Seminar on Stochastic Analysis, Random Fields and Applications, III (Ascona, 1999) 269–291, Progr. Probab., 52, Birkh¨ auser, Basel, 2002. [24] Walsh, J. B.: An introduction to stochastic partial differential equations; In: Ecole d’Et´ e de Probabilit´ es de Saint Flour XIV, Lecture Notes in Math. 1180 (1986) 265-438. Oana Mocioalca: Department of Mathematical Sciences, Kent State University, P.O. Box 5190, Kent, OH, 44242, USA E-mail address: [email protected] URL: http://www.math.kent.edu/∼oana Frederi Viens: Department of Statistics, Purdue University, 150 N. University St., West Lafayette, IN 47907-2067, USA E-mail address: [email protected] URL: http://www.stat.purdue.edu/∼viens


QUANTUM STOCHASTIC PROCESS ASSOCIATED WITH ´ QUANTUM LEVY LAPLACIAN ˆ UN CIG JI*, HABIB OUERDIANE, AND KIMIAKI SAITO** Abstract. A noncommutative extension of the Lévy Laplacian, called the quantum Lévy Laplacian, is introduced and its relevant properties are studied, in particular, a relation between the classical and quantum Lévy Laplacians is studied. We construct a semigroup and a quantum stochastic process generated by the quantum Lévy Laplacian.

1. Introduction Since an infinite dimensional analogue of the usual Laplacian on an Euclidean space has been introduced and studied by Lévy in his famous book [15] and so called the Lévy Laplacian, the Lévy Laplacian has been studied in [5, 8, 21]. On the other hand, the Lévy Laplacian acting on white noise functionals has been introduced by Hida and studied by many authors within the framework of white noise theory, see [6, 11, 12] and the references cited therein. In recent years the Lévy Laplacian has afforded us much interest for its newly discovered relations with certain stochastic processes [2, 22, 25], Yang–Mills equations [3, 14], Gross Laplacian [9, 12], infinite dimensional rotation group [16], quadratic quantum white noise [19, 20], Poisson noise functionals [24] and so on. In recent papers [1, 4, 5, 10], noncommutative generalizations of the Lévy Laplacian, called the quantum Lévy Laplacian, acting on operators have been introduced and studied. In particular, in [1], the authors studied the quantum extension of the time shift of the Brownian motion to give a positive answer to the Meyer’s problem. Then the generator of the Markov semigroup generated by the quantum extension of the time shift is a quantum Laplacian. If we consider the Ces` aro Hilbert space as a state space, then the generator is called the quantum Lévy Laplacian. In this paper, we introduce a new type of noncommutative generalization of the Lévy Laplacian as follows: Consider a space L((S), (S)∗ ) of white noise operators within white noise Gelfand triple (S) ⊂ (L2 ) ⊂ (S)∗ . Then by the kernel theorem there is a topological isomorphism U : L((S), (S)∗ ) −→ (S)∗ ⊗ (S)∗ 2000 Mathematics Subject Classification. Primary 60H40; Secondary 46A32, 46F25, 60G52, 81S25. Key words and phrases. white noise operator, operator symbol, quantum Lévy Laplacian, quantum stochastic process. * Supported in part by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD)”(No. R05-2004-000-11346-0). ** Supported in part by JSPS Grant-in-Aid for Scientific Research (C) No. 17540136. 231

232

ˆ UN CIG JI, HABIB OUERDIANE, AND KIMIAKI SAITO

since (S) is a nuclear space. The classical Lévy Laplacian ∆L acting on generalized white noise functionals is extended, denoted by ∆L ⊗ I + I ⊗ ∆L , to a subspace of (S)∗ ⊗ (S)∗ as a Lévy Laplacian acting on white noise functionals in two variables, where I is the identity operator on (S)∗ . Then for Ξ belonging to a certain class of white noise operators, the quantum Lévy Laplacian ∆QL is given by ∆QL Ξ = U −1 (∆L ⊗ I + I ⊗ ∆L ) UΞ using the topological isomorphism U which can be easily understood through the (classical) Lévy Laplacian and the kernel theorem in the white noise theory. Therefore, the difficulty of domain problem for the quantum Lévy Laplacian comes back to the problem for the (classical) Lévy Laplacian. This point of view is one of our main purpose of this paper. This representation implies a natural quantumclassical correspondence (Theorem 4.8). Moreover, the quantum Lévy Laplacian in [1] can be represented by the quantum Lévy Laplacian in our approach. Note that the noncommutative extension of the Lévy Laplacian introduced by Ji–Obata– Ouerdiane [10] is slightly different from our quantum Lévy Laplacian. In [10], the quantum Lévy Laplacian has been introduced by using the Wick symbols of white noise operators. On the other hand, in this paper we use the symbols of white noise operators for the quantum Lévy Laplacian. We study relevant properties of the quantum Lévy Laplacian, and construct a semigroup and a quantum stochastic process generated by the quantum Lévy Laplacian. The paper is organized as follows. In Section 2 we recall the spaces of white noise functionals. In Section 3 we remind the theory of white noise operators with study of the basic topological isomorphism U. In Section 4 we introduce the quantum Lévy Laplacian and study its properties with the natural quantumclassical correspondence. In Section 5 we study a domain of the quantum Lévy Laplacian. In Section 6 we construct an one parameter semigroup generated by the quantum Lévy Laplacian. In Section 7 we construct a quantum stochastic process associated with the quantum Lévy Laplacian. 2. White Noise Functionals Let HR = L2R (R, dt) be the real Hilbert space of square integrable functions with respect to the Lebesgue measure dt and let A = 1 + t2 − d2 /dt2 be the harmonic oscillator. The norm on HR is denoted by | · |0 . Then for each p ≥ 0, Sp,R = {ξ ∈ HR ; | ξ |p ≡ | Ap ξ |0 < ∞} becomes a Hilbert space and we have a Gelfand triple: SR = proj lim Sp,R ⊂ HR ⊂ S ∗ ∼ = ind lim S−p,R , R

p→∞

p→∞

where ∼ = stands the topological isomorphism and for each p > 0, S−p,R is the completion of the Hilbert space HR with respect to the Hilbertian norm | · |−p = | A−p · |0 . Then SR becomes a nuclear space and coincides with the Schwartz space. The complexification of a real locally convex space XR is denoted by XC . If there is no confusion, we use the symbol X for XC . The Boson Fock space over the (complex) Hilbert space H ≡ HC , denoted by b ⊗n Γ(H), is the Hilbert space consisting of sequences (fn )∞ (the n=0 , where fn ∈ H

´ QUANTUM LEVY LAPLACIAN

n-fold symmetric tensor product of H) and norm on H ⊗n ∼ = L2C (Rn ) for each n. For p ∈ R, we put k φ k2p =

∞ X

n! | fn |2p ,

P∞

n=0

233 2

n! | fn |0 < ∞, where | · |0 is the

φ = (fn ) ∈ Γ(H),

n=0

where | fn |p = | (A⊗n )p fn |0 . We set n o (S) = φ ∈ Γ(H) ; k φ kp < ∞ for all p ∈ R

which becomes a countable Hilbert nuclear space with the topology induced from the norms {k · kp ; p ∈ R}. We thus come to a complex Gelfand triple: (S) ⊂ Γ(H) ⊂ (S)∗ . Let µ be the standard Gaussian measure on SR∗ of which the characteristic functional is given by Z 1 2 ξ ∈ SR . exp{i hx, ξi}dµ(x) = exp − |ξ|0 , 2 SR∗ Let (L2 ) ≡ L2 (SR∗ , µ) be the Hilbert space of C-valued square integrable functions on SR∗ . By the Wiener-Itˆ o decomposition theorem, each φ ∈ (L2 ) admits an expression ∞ X

⊗n φ(x) = : x :, fn , x ∈ SR∗ , (2.1) n=0

b ⊗n

⊗n

and : x : denotes the Wick ordering of x⊗n . Moreover, the where fn ∈ H 2 (L )-norm k φ k0 of φ is given by 2 k φ k0

=

∞ X

n!|fn |20 .

n=0

An element of (L2 ) is called a white noise functional. The nuclear space of white noise functionals which is corresponding to (S) under the Wiener-Itˆ o-Segal isomorphism is denoted by the same symbol (S). The strong dual space of the nuclear space (S) is also denoted by the same symbol (S)∗ . An element of (S)∗ is called a generalized white noise functionals. Thus we have the following Gelfand triple: (S) ⊂ (L2 ) ⊂ (S)∗ which is referred to as the Hida–Kubo–Takenaka space (see [11, 18]). For each ⊗n ∗ Φ ∈ (S)∗ , there exists a unique sequence {Fn }∞ )sym such that n=0 , Fn ∈ (S hhΦ, φii =

∞ X

n! hFn , fn i

n=0

for all φ ∈ (S) given as in (2.1). Moreover, k Φ k2−p

=

∞ X

n=0

n!|Fn |2−p < ∞

234


for some p ≥ 0. In this case, we use a formal expression for Φ ∈ (S)∗ : Φ(x) =

∞ X

n=0

: x⊗n :, Fn ,

x ∈ SR∗

or

Φ = (Fn )∞ n=0 .

For p ≥ 0 let (S)p be the space of all φ ∈ (L2 ) such that ||φ||p < ∞ and (S)−p the completion of (L2 ) with respect to the norm || · ||−p . Then, for each p ∈ R, equipped with the norm || · ||p , (S)p becomes a Hilbert space. The space is identified with the inductive limit space (S)∗ = ind lim (S)−p . p→∞

For each ξ ∈ S ≡ SC , an exponential vector (or coherent vector ) φξ is defined by φξ (x) =

∞ X

n=0

: x⊗n :,

ξ ⊗n n!

1 = exp hx, ξi − hξ, ξi 2

which is corresponding to (ξ ⊗n /n!)∞ n=0 denoted by the same symbol φξ . Then for each z ∈ S ∗ , the exponential vector φz is defined as a generalized white noise functional corresponding to (z ⊗n /n!)∞ n=0 . It is well-known that {φξ ; ξ ∈ S} spans a dense subspace of (S) and then white noise functional Φ ∈ (S)∗ is uniquely specified by the S-transform SΦ of Φ defined by SΦ(ξ) = hhΦ, φξ ii ,

ξ ∈ S.

3. White Noise Operators Since every continuous linear operator from (S) into (S)∗ has a Fock expansion [18] which can be considered as a superposition of the quantum white noise, a continuous linear operator from (S) into (S)∗ is called a white noise operator. The space of all white noise operators is denoted by L((S), (S)∗ ) equipped with the bounded convergence topology. A white noise operator Ξ is uniquely specified by its symbol which is a C–valued function on S × S defined by b η) = hhΞφξ , φη ii , Ξ(ξ,

ξ, η ∈ S.

b of a white noise operator It is straightforward to see that the symbol Θ = Ξ ∗ Ξ ∈ L((S), (S) ) possesses the following properties: (O1) for any ξ, ξ1 , η, η1 ∈ S the function (z, w) 7→ Θ(zξ + ξ1 , wη + η1 ) is entire holomorphic on C × C; (O2) there exist constant numbers C ≥ 0 and p ≥ 0 such that ξ, η ∈ S. (3.1) |Θ(ξ, η)| ≤ K exp |ξ|2p + |η|2p ,

The following characterization of symbols is proved by Obata in [17].

Theorem 3.1. Let Θ be a C–valued function defined on S × S. Then Θ is the symbol of some Ξ ∈ L((S), (S)∗ ) if and only if Θ satisfies the conditions (O1) and (O2).


235

For each x ∈ S ∗ the annihilation operator a(x) ∈ L((S), (S)∗ ) is defined by a(x)φξ = hx, ξi φξ ,

ξ ∈ S.

and the adjoint a∗ (x) is called the creation operator. Note that a(x) ∈ L((S), (S)) and a∗ (x) ∈ L((S)∗ , (S)∗ ). Since (S) is a nuclear space, by the kernel theorem we have the following isomorphism: L((S), (S)∗ ) ∼ = (S)∗ ⊗ (S)∗ , ∗ i.e., for each Ξ ∈ L((S), (S) ) there exists a unique ΦΞ ∈ (S)∗ ⊗ (S)∗ such that hhΞφ, ϕii = hhΦΞ , ϕ ⊗ φii , ∗

φ, ϕ ∈ (S). ∗

Define a map U : L((S), (S) ) 3 Ξ 7→ ΦΞ ∈ (S) ⊗ (S)∗ . Conversely, for each φx1 ⊗ φx2 ∈ (S)∗ ⊗ (S)∗ , there exists a unique Ξφx1 ⊗φx2 ∈ L((S), (S)∗ ) such that

Ξφx1 ⊗φx2 φ, ϕ = hhφx1 ⊗ φx2 , ϕ ⊗ φii = hhφx1 , ϕii hhφx2 , φii , φ, ϕ ∈ (S).

Therefore, for any x1 , x2 ∈ S ∗ and ξ, η ∈ S we have ∞ X 1 ⊗l ⊗l b φx ⊗φx (ξ, η) = ehx1 , ηi+hx2 , ξi = Ξ x ⊗ x⊗m ⊗ ξ ⊗m . 2 , η 2 1 l!m! 1

(3.2)

l,m=0

4. Classical and Quantum L´ evy Laplacians

4.1. Classical L´ evy Laplacian. Let F ∈ C 2 (S). Then for each ξ ∈ S there 0 ∗ exist F (ξ) ∈ S and F 00 (ξ) ∈ (S ⊗ S)∗ such that 1 2 η∈S F (ξ + η) = F (ξ) + hF 0 (ξ), ηi + hF 00 (ξ), η ⊗ ηi + o(| η |p ), 2 for some p ≥ 0. Moreover, both maps ξ 7→ F 0 (ξ) ∈ S ∗ and ξ 7→ F 00 (ξ) ∈ (S ⊗ S)∗ are continuous. For more study, we refer to [7]. Now suppose we are given an infinite sequence {en }∞ n=1 ⊂ S. The sequence {en }∞ n=1 is not necessarily orthogonal, but in order to ensure some reasonable properties of the Lévy Laplacian, we need to assume some conditions on {en }. We shall do so afterwards, see also the original definition due to Lévy [15]. We consider the Ces` aro mean of hF 00 (ξ), en ⊗ en i defined by N 1 X 00 hF (ξ), en ⊗ en i. N →∞ N n=1

(4.1)

lim

The limit does not necessarily exist. Moreover, the limit depends not only on the choice of the sequence {en } but also its arrangement. Let DL ≡ DL (S, {en }) be the subspace of F ∈ C 2 (S) for which the limit (4.1) exists for all ξ ∈ S. For F ∈ DL we define N 1 X 00 hF (ξ), en ⊗ en i , N →∞ N n=1

∆L F (ξ) = lim

ξ ∈ S.

The operator ∆L is called the Lévy Laplacian on S associated with {en }. 2 We fix a finite interval T on R. Take an orthogonal basis {en }∞ n=1 ⊂ S for L (T ) satisfying the equally dense and uniform boundedness property (see [11, 12, 16]).


236

Let DL ≡ Dom (∆L ) denote the domain of the Lévy Laplacian ∆L consisting of all Φ ∈ (S)∗ satisfying that S(Φ) ∈ DL and there exists ΨΦ ∈ (S)∗ such that N 1 X S(Φ)00 (ξ)(en , en ) S(ΨΦ )(ξ) = lim N →∞ N n=1

Then the Lévy Laplacian ∆L is defined by

Φ ∈ DL ⊂ (S)∗ .

∆L Φ = Ψ Φ ,

Let DTL denote the set of all functionals Φ ∈ DL such that S(Φ)(η) = 0 for all η ∈ S with supp(η) ⊂ T C . 4.2. Quantum L´ evy Laplacian. Let F ∈ C 2 (S × S). Then for each ξ1 , ξ2 ∈ S 0 there exist Fi (ξ1 , ξ2 ) ∈ S ∗ and Fij00 (ξ1 , ξ2 ) ∈ (S ⊗ S)∗ , i, j = 1, 2, such that F (ξ1 + η1 , ξ2 + η2 )

= F (ξ1 , ξ2 ) +

2 X

hFi0 (ξ1 , ξ2 ), ηi i

i=1

+

1 2

2 X

i,j=1

2 2 Fij00 (ξ1 , ξ2 )ηi , ηj + o(| η1 |p + | η2 |p )

for some p ≥ 0 and any η1 , η2 ∈ S. Let DLQ be the subspace of F ∈ C 2 (S × S) for which the limits N E 1 X D b 00 Ξii (ξ, η), en ⊗ en , N →∞ N n=1

lim

i = 1, 2

exists for all ξ, η ∈ S. Consider the set consisting of Ξ ∈ L((S), (S)∗ ) for which b ∈ DQ and there exists ΥΞ ∈ L((S), (S)∗ ) such that Ξ L 2 X N D E X b Ξ (ξ, η) = lim 1 b 00 (ξ, η), en ⊗ en , Υ Ξ ii N →∞ N i=1 n=1

ξ, η ∈ S.

(4.2)

This set serves as the domain of the operator ∆QL in the next definition and will be denoted by Dom (∆QL ). Definition 4.1. For each Ξ ∈ Dom (∆QL ), we write ∆QL Ξ ≡ ΥΞ ,

(4.3)

where ΥΞ is given as in (4.2). Then ∆QL is called the quantum Lévy Laplacian. Let SL∗ be the set of all elements f ∈ S ∗ such that the limit N 1 X 2 hf, en i N →∞ N n=1

hf, f iL = lim exists.

Proposition 4.2. For any f, g ∈ SL∗ , Ξφf ⊗φg ∈ Dom (∆QL ) and we have ∆QL Ξφf ⊗φg = (hf, f iL + hg, giL ) Ξφf ⊗φg . Proof. The proof is straightforward from (3.2).


237

Lemma 4.3 ([10]). For η, ζ ∈ S and z ∈ C we have φη+zζ =

∞ X zn ∗ n a (ζ) φη , n! n=0

where the right hand side converges in (S) uniformly in z running over a compact set in C. Therefore, dn φη+zζ = a∗ (ζ)n φη dz n z=0 holds in (S).

Lemma 4.4. Let ζ ∈ S and Ξ ∈ L((S), (S)∗ ). Then d2 \ 2 Ξ(ξ, η) = b η + zζ), a(ζ) Ξ(ξ, dz 2 z=0 d2 \ ∗ (ζ)2 (ξ, η) = b + zζ, η). Ξ(ξ Ξa dz 2 z=0

∗

Proof. For any ζ ∈ S and Ξ ∈ L((S), (S) ), by applying Lemma 4.3 we have

d2 \ 2 Ξ(ξ, η) = φ a(ζ) Ξφξ , a∗ (ζ)2 φη = Ξφξ , η+zζ dz 2 z=0

which follows the first identity from the continuity of the canonical bilinear form hh·, ·ii. Similarly, the second identity is proved. Theorem 4.5. For each Ξ ∈ Dom (∆QL ) we have N 1 X hh(a(en )2 Ξ + Ξa∗ (en )2 )φξ , φη ii, N →∞ N n=1

hh(∆QL Ξ)φξ , φη ii = lim

ξ, η ∈ S. (4.4)

Proof. By Lemma 4.4 we see that

d2 \ 2 b η + zen ), Ξ(ξ, hh(a(en ) Ξ)φξ , φη ii = a(en ) Ξ(ξ, η) = dz 2 z=0 d2 ∗ 2 2 \ ∗ b + zen , η). hhΞ(aen ) φξ , φη ii = Ξ(aen ) (ξ, η) = Ξ(ξ dz 2 z=0 2

If Ξ ∈ Dom (∆QL ), the limit N 2 1 X d2 b η + zen ) + d b + zen , η) lim Ξ(ξ, Ξ(ξ N →∞ N dz 2 z=0 dz 2 z=0 n=1 \ exists and coincides with ∆ QL Ξ(ξ, η). Hence

N 1 X \ hh(a(en )2 Ξ + Ξa∗ (en )2 )φξ , φη ii = ∆ QL Ξ(ξ, η), N →∞ N n=1

lim

from which (4.4) follows.

238


Theorem 4.6. For each Ξ ∈ Dom (∆QL ), we have hh(∆QL Ξ)φξ , φη ii =

N 1 X −1 hh U I ⊗ a(en )2 UΞ φξ , φη ii, N →∞ N n=1

lim

N 1 X −1 hh U a(en )2 ⊗ I UΞ φξ , φη ii, N →∞ N n=1

+ lim

ξ, η ∈ S.

Proof. For any ξ, η ∈ S, we have hhΞa∗ (en )2 φξ , φη ii

= hhUΞ, φη ⊗ a∗ (en )2 φξ ii = hh I⊗a(en )2 UΞ, φη ⊗ φξ ii = hh U −1 I ⊗ a(en )2 UΞ φξ , φη ii.

Similarly, we prove that for any ξ, η ∈ S hha(en )2 Ξφξ , φη ii = hh U −1 a(en )2 ⊗ I UΞ φξ , φη ii. Then the proof is completed by applying Theorem 4.5.

Remark 4.7. It is well–known in various contexts that N 1 X a(en )2 N →∞ N n=1

∆L = lim

on Dom (∆L ).

In fact, the right hand side is given a meaning as following: for Φ ∈ Dom (∆L ) we have N 1 X hh∆L Φ, φξ ii = lim hha(en )2 Φ, φξ ii, ξ ∈ S. (4.5) N →∞ N n=1 Therefore, from Theorem 4.6 we can write

∆QL = U −1 (I ⊗ ∆L ) U + U −1 (∆L ⊗ I) U.

(4.6)

4.3. Quantum–Classical Correspondence. For each φ, ψ ∈ (S), we write φψ for the pointwise multiplication. It is well-known that the pointwise multiplication yields a continuous bilinear map from (S)×(S) into (S), see [11, 18]. For Φ ∈ (S)∗ and φ ∈ (S) we define Φφ = φΦ ∈ (S)∗ by hhΦφ, ψii = hhΦ, φψii ,

ψ ∈ (S).

Obviously, the map (Φ, φ) 7→ Φφ is a separately continuous bilinear map. In particular, each Φ ∈ (S)∗ gives rise to a multiplication operator MΦ ∈ L((S), (S)∗ ) ∼ = (S)∗ ⊗ (S)∗ defined by MΦ φ = Φφ. With this we have a continuous injection (S)∗ ,→ L((S), (S)∗ ). Note also that (MΦ )∗ = MΦ . Theorem 4.8. Let Φ ∈ Dom (∆L ). Then MΦ ∈ Dom (∆QL ) and 1 (∆QL MΦ ) φ0 = ∆L Φ, 2 where φ0 is the vacuum vector.

(4.7)


239

Proof. For any ξ, η ∈ S, by the derivation property of a(ζ) we have N 1 X hha(en )2 (Φφξ ) , φη ii N →∞ N n=1

lim

N 1 X hh a(en )2 Φ φξ , φη ii N →∞ N n=1

= lim

N 1 X 2hh(a(en )Φ) (a(en )φξ ) , φη ii + hhΦ a(en )2 φξ , φη ii . N →∞ N n=1

+ lim

On the other hand, we can easily see that

N 1 X 2hh(a(en )Φ) (a(en )φξ ) , φη ii + hhΦ a(en )2 φξ , φη ii = 0. N →∞ N n=1

lim

Therefore, for any ξ, η ∈ S we have

N N 1 X 1 X 2 lim hha(en ) (Φφξ ) , φη ii = lim hh a(en )2 Φ φξ , φη ii N →∞ N N →∞ N n=1 n=1

Similarly, we prove that for any ξ, η ∈ S

N N 1 X 1 X hhΦa∗ (en )2 φξ , φη ii = lim hh a(en )2 Φ φξ , φη ii. N →∞ N N →∞ N n=1 n=1

lim

Therefore, by applying Theorem 4.5 with Ξ = MΦ , MΦ ∈ Dom (∆QL ) and we have hh(∆QL MΦ )φ0 , φη ii = 2 hh∆L Φ, φη ii ,

η∈S

which implies (4.7).

(4.8)

5. New Domains of L´ evy Laplacians b b Given f ∈ L1C (R)⊗l ∩ L2C (R)⊗l and αk ∈ R, k = 1, 2, · · · , l, we consider a generalized white noise functional Φ in DTL of the form: Z Φ= f (s1 , · · · , sl ) : eiα1 x(s1 ) · · · eiαl x(sl ) : ds, (5.1) Tl

where : · : is the Wick ordering. The S-transform of Φ is given by Z S(Φ)(ξ) = f (s1 , · · · , sl )eiα1 ξ(s1 ) · · · eiαl ξ(sl ) ds. Tl

By direct computation we can see that l

∆L Φ =

1 X 2 αk − |T | k=1

!

Φ.

We fix a continuous function γ : R → [0, ∞) such that there exists a stochastic process {Xt ; t ≥ 0} with E[eizXt ] = e−tγ(z) for each t ≥ 0. For each n ∈ N, let Dγn


240

be the linear space spanned by generalized white noise functionals of the form in Eq. (5.1), i.e., Z n Y Φ= f (s) : eiαk x(sk ) : ds, Tn

where f ∈ condition:

b L1C (R)⊗n

∩

b L2C (R)⊗n

α1 + · · · + α n =

p

k=1

and αk ∈ R \ {0}, k = 1, 2, . . . , n satisfy the α21 + · · · + α2n = |T |γ(n).

|T |n,

We also put Dγ0 = C. Then Dγn is a linear subspace of (S)−p for any p > [13]) and ∆L is a linear operator from Dγn into itself such that

5 12

(see

Φ ∈ Dγn .

k∆L Φk−p = γ(n)kΦk−p , γ

Let Dn be the completion of Dγn in (S)−p with respect to k · k−p . Then for each γ n ∈ N ∪ {0}, Dn becomes a Hilbert space with the inner product of (S)−p . For each n ∈ N ∪ {0}, the operator ∆L can be extended to a continuous linear operator γ ∆L from Dn into itself satisfying γ

k∆L Φk−p = γ(n)kΦk−p ,

Φ ∈ Dn . γ

The operator ∆L is a self-adjoint operator on Dn for each n ∈ N ∪ {0}. P∞ P∞ Proposition 5.1 ([23]). Let Φ = n=0 Φn , Ψ = n=0 Ψn be generalized white γ noise functionals such that Φn , Ψn ∈ Dn for each n ∈ N ∪ {0}. If Φ = Ψ in (S)∗ , then Φn = Ψn for each n ∈ N ∪ {0}. 5 From now on, p > 12 will be an arbitrarily fixed number. For each N ∈ N∪{0}, we put ) (∞ N X ∞

X X γ

k 2 γ ∗ Φn ∈ (S) ; S−p,N =

∆L Φn < ∞, Φn ∈ Dn , n = 0, 1, 2, · · · . n=0

−p

k=0 n=0

By the Schwarz inequality we see that for any N ∈ N, Sγ−p,N is contained in (S)−p and is a Hilbert space equipped with the new norm ||| · |||−p,N given by 2 ||| Φ |||−p,N

N X ∞

X

k 2 =

∆L Φn , −p

k=0 n=0

Φ=

∞ X

Φn ∈ Sγ−p,N .

n=0

Moreover, in view of the inclusion relations:

· · · ⊂ · · · ⊂ Sγ−p,N +1 ⊂ Sγ−p,N ⊂ · · · ⊂ Sγ−p,1 ⊂ (S)−p , we define Sγ−p,∞ = proj lim Sγ−p,N = N →∞

Sγ−p,∞

γ Dn

∞ \

Sγ−p,N .

N =1

for any n ∈ N ∪ {0}. The operator ∆L can Note that the space includes be extended to a continuous linear operator from Sγ−p,2 into Sγ−p,1 , denoted by the same notation ∆L , satisfying ∆L Φ ≤ ||| Φ ||| , Φ ∈ Sγ , N = 1, 2, . . . . −p,N

−p,N +1

−p,∞


241

Therefore, ∆L is a continuous linear operator from Sγ−p,∞ into itself. Theorem 5.2 ([23]). The operator ∆L is a self-adjoint operator densely defined on Sγ−p,N for each N ∈ N. Put

Sγ−p,∞ ⊗ Sγ−p,∞ = proj lim Sγ−p,N ⊗ Sγ−p,N ⊂ (S)−p ⊗ (S)−p N →∞

and

Lγ−p,∞ = U −1 Sγ−p,∞ ⊗ Sγ−p,∞ . For any Φ, Ψ ∈ Dγn , by (4.6) we have U∆QL U −1 Φ ⊗ Ψ = (∆L ⊗ I + I ⊗ ∆L ) Φ ⊗ Ψ. Therefore, the following result is straightforward.

Theorem 5.3. The quantum Lévy Laplacian can be extended to Lγ−p,∞ as a continuous linear operator, denoted by the notation ∆QL . In this case, we have ∆QL = U −1 ∆L ⊗ I + I ⊗ ∆L U. From now on we write

LQ = ∆ L ⊗ I + I ⊗ ∆ L . Then the following result is straightforward from Theorem 5.2. Theorem 5.4. The operator LQ is a continuous linear operator from Sγ−p,∞ ⊗ Sγ−p,∞ into itself. Moreover, LQ is a self-adjoint operator densely defined on Sγ−p,N ⊗ Sγ−p,N for each N ∈ N. 6. Semigroups Generated by L´ evy Laplacians For each t ≥ 0 we define Gt on Sγ−p,∞ by Gt Φ =

∞ X

e−γ(n)t Φn ,

Φ=

n=0

∞ X

Φn ∈ Sγ−p,∞ ,

γ

Φn ∈ Dn , n = 0, 1, 2, · · · .

n=0

Then for any N ∈ N we have 2 ||| Gt Φ |||−p,N

=

N X ∞ X

2

γ(n)2k e−2γ(n)t k Φn k−p

k=1 n=0

≤ ||| Φ |||2−p,N . Therefore, {Gt }t≥0 is equi-continuous in t. On the other hand, for each t, t0 ≥ 0 and any N ∈ N, we obtain that N X ∞ 2 X γ(n)2k e−γ(n)t − e−γ(n)t0 k Φn k2−p ||| Gt Φ − Gt0 Φ |||2−p,N = k=1 n=0 2

≤ 4 || Φ |||−p,N .

Therefore, by the Lebesgue dominated convergence theorem, we see that lim Gt Φ = Gt0 Φ in S−p,∞ .

t→t0


242

It is easily checked the semigroup properties that G0 = I and Gt Gs = Gt+s for any s, t ≥ 0. Hence {Gt }t≥0 is an equi-continuous C0 -semigroup. For each t ≥ 0, we put GtF = Gt ⊗ Gt on Sγ−p,∞ ⊗ Sγ−p,∞ . Then it is straightforward that {GtF }t≥0 is an equi-continuous C0 -semigroup on Sγ−p,∞ ⊗ Sγ−p,∞ . Proposition 6.1. The infinitesimal generator of the semigroup {GtF }t≥0 is LQ . Proof. With the help of Theorem 5.4, the proof is a simple modification of the proof of Proposition 5 in [23]. For each t ≥ 0 we put GtQ = U −1 GtF U. Then {GtQ }t≥0 becomes a semigroup on Lγ−p,∞ and the following result is obvious from Proposition 6.1. Theorem 6.2. The infinitesimal generator of the semigroup {GtQ }t≥0 is ∆QL . 7. Stochastic Processes Generated by L´ evy Laplacians Let {X1,t }t≥0 and {X2,t }t≥0 be independent continuous stochastic processes of which the characteristic functions of X1,t and X2,t are given by E eizX1,t = E eizX2,t = e−tγ(z)

and let ζT be a smooth function in S with ζT (u) = (1/|T |)1/2 on T . We use the γ \ b =∆ same notation ∆QL as ∆QL (Ξ) QL Ξ for any Ξ ∈ L−p,∞ . Then we can construct a quantum stochastic process generated by the quantum Lévy Laplacian as follows. Theorem 7.1. Let Θ be the symbol of a white noise operator in Lγ−p,∞ . Then it holds that et∆QL Θ(ξ, η) = E [Θ(ξ + X1,t ζT , η + X2,t ζT )] ,

ξ, η ∈ S.

Proof. Let l, m ∈ N. Suppose that Θ is given by Z l m Y Y iαj ξ(sj ) Θ(ξ, η) = f (s1 , · · · , sl ; t1 , · · · , tm ) e eiβk η(tk ) dsdt (7.1) T l+m

b ⊗(l+m)

j=1

k=1

b ⊗(l+m)

with f ∈ L1C (R) ∩ L2C (R) , and αj ∈ R \ {0}, j = 1, 2, . . . , l, βk ∈ R \ {0}, k = 1, 2, . . . , m satisfying the condition: p α1 + · · · + αl = |T |l, α21 + · · · + α2l = |T |γ(l), p 2 β12 + · · · + βm β1 + · · · + βm = |T |m, = |T |γ(m). (7.2) Then we have

E [Θ(ξ + X1,t ζT , η + X2,t ζT )] = e−t(γ(l)+γ(m)) Θ(ξ, η) = et∆QL Θ(ξ, η), ξ, η ∈ S.


243

P∞ γ γ Let Θ = l,m=0 Θl,m belong to the set L\ −p,∞ of all symbols of operators in L−p,∞ . Denote the integral given as in (7.1) by Iα,β (f )(ξ, η). Then for each l, m ∈ N ∪ {0}, Θl,m can be expressed as the form: X Iα[N ] ,β [N ] (fα[N ] ;β [N ] )(ξ, η) Θl,m (ξ, η) = lim N →∞

α[N ] ,β [N ]

b b for a sequence fα[N ];β [N ] , N = 1, 2, . . . of functions in L1C (R)⊗(l+m) ∩ L2C (R)⊗(l+m) , X [N ] [N ] [N ] where means a sum of finitely many terms on α = (α1 , . . . , αl ) ∈ α[N ] ,β [N ] l

[N ]

[N ]

(R \ {0}) and β [N ] = (β1 , . . . , βm ) ∈ (R \ {0})m satisfying (7.2). Therefore, we have ∞ X E [|Θl,m (ξ + X1,t ζT , η + X2,t ζT )|] l,m=0

 X ilX1,t imX2,t   Iα[N ] ,β [N ] (fα[N ] ;β [N ] )(ξ, η)e e = E lim N →∞ l,m=0 α[N ] ,β [N ] ∞ X X = lim Iα[N ] ,β [N ] (fα[N ] ;β [N ] )(ξ, η) N →∞ l,m=0 α[N ] ,β [N ] ∞ X

=

∞ X



|Θl,m (ξ, η)| .

l,m=0 γ On the other hand, Θl,m = Ξd l,m for some Ξl,m ∈ L−p,∞ and so ∞ X

l,m=0

|Θl,m (ξ, η)| ≤

∞ X

kUΞl,m k−p kφξ kp kφη kp < ∞,

ξ, η ∈ S.

l,m=0

Therefore, by continuity of et∆QL we obtain that E [Θ(ξ + X1,t ζT , η + X2,t ζT )]

= =

∞ X

l,m=0 ∞ X

E [Θl,m (ξ + X1,t ζT , η + X2,t ζT )] et∆QL Θl,m (ξ, η)

l,m=0

= et∆QL Θ(ξ, η) which proves the assertion.

For each Φ ∈ (S)∗ and η ∈ S, the translation Tη Φ of Φ is well defined as an element in (S)∗ by Tη Φ(·) = Φ(· + η) and then S(Tη Φ)(ξ) = S(Φ)(ξ + η) for any ξ ∈ S. Therefore, for any η, ζ ∈ S and any operator Ξ ∈ L((S), (S)∗ ) we define Tη,ζ Ξ by Tη,ζ Ξ = U −1 (Tη ⊗ Tζ ) UΞ. With this notation the following corollary is obvious from Theorem 7.1.

244


Corollary 7.2. Let Ξ ∈ Lγ−p,∞ . Then it holds that GtQ Ξ = et∆QL Ξ = E TX1,t ζT ,X2,t ζT Ξ . Corollary 7.3. Let Φ ∈ Sγ−p,∞ . Then it holds that Gt Φ = et∆L Φ = E TX1,t ζT Φ .

Proof. For each Φ ∈

Sγ−p,∞ ,

ΞΦ⊗φ0 = U

−1

(Φ ⊗ φ0 ) ∈

Lγ−p,∞

(7.3) and

GtQ (ΞΦ⊗φ0 ) φ0 = (ΞGt Φ⊗φ0 ) φ0 = Gt Φ. and, by applying Corollary 7.2 and Theorem 5.3, we have GtQ (ΞΦ⊗φ0 ) φ0 = et∆QL (ΞΦ⊗φ0 ) φ0 = Ξet∆L Φ⊗φ0 φ0 = et∆L Φ which proves the first identity in (7.3). On the other hand, for any η, ζ ∈ S Tη,ζ ΞΦ⊗φ0 = ΞTη Φ⊗φ0 , Therefore, for any η ∈ S

E TX1,t ζT ,X2,t ζT ΞΦ⊗φ0 φ0 , φη which implies that

(Tη,ζ ΞΦ⊗φ0 ) φ0 = Tη Φ.

TX1,t ζT ,X2,t ζT ΞΦ⊗φ0 φ0 , φη

= E TX1,t ζT Φ, φη

= E TX1,t ζT Φ , φη

= E

E TX1,t ζT ,X2,t ζT ΞΦ⊗φ0 φ0 = E TX1,t ζT Φ .

Therefore, by Corollary 7.2 we have GtQ ΞΦ⊗φ0 φ0 = E TX1,t ζT ,X2,t ζT ΞΦ⊗φ0 φ0 = E TX1,t ζT Φ which proves the second identity in (7.3).

The results in this paper can be extended to a more general Gelfand triple if we replace the test function (S) by a test function space W = Fθ (N ) as in [10]. References 1. Accardi, L., Barhoumi, A. and Ouerdiane, H.: A quantum approach to Laplace operators; Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9 (2006) 215–248. 2. Accardi, L. and Bogachëv, V.: The Ornstein–Uhlenbeck process associated with the Lévy Laplacian and its Dirichlet form; Prob. Math. Stat. 17 (1997) 95–114. 3. Accardi, L., Gibilisco, P. and Volovich, I. V.: Yang-Mills gauge fields as harmonic functions for the Lévy Laplacian; Russian J. Math. Phys. 2 (1994) 235–250. 4. Accardi, L., Ouerdiane, H. and Smolyanov, O. G.: Lévy Laplacian acting on operators; Russian J. Math. Phys. 10 (2003) 359–380. 5. Accardi, L. and Smolyanov, O. G.: Lévy-Laplace operators in functional rigged Hilbert spaces; Math. Notes 72 (2002) 129–134. 6. Chung, D. M., Ji, U. C. and Saitˆ o, K.: Cauchy problems associated with the Lévy Laplacian in white noise analysis; Infin. Dimens. Anal. Quantum Probab. Rel. Top. 2 (1999) 131–153. 7. Dineen, S.: Complex Analysis on Infinite Dimensional Spaces. Springer-Verlag, 1999. 8. Feller, M. N.: Infinite-dimensional elliptic equations and operators of Lévy type; Russian Math. Surveys 41 (1986) 119–170.


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9. Ji, U. C. and Saitˆ o, K.: A similarity between the Gross Laplacian and the Lévy Laplacian; Preprint, 2006. 10. Ji, U. C., Obata, N. and Ouerdiane, H.: Quantum Lévy Laplacian and associated heat equation; Preprint, 2005. 11. Kuo, H.-H.: White Noise Distribution Theory. CRC Press, 1996. 12. Kuo, H.-H., Obata, N. and Saitˆ o, K.: Lévy Laplacian of generalized functions on a nuclear space; J. Funct. Anal. 94 (1990) 74–92. 13. Kuo, H.-H., Obata, N. and Saitˆ o, K.: Diagonalization of the Lévy Laplacian and related stable processes; Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5 (2002) 317–331. 14. Leandre, R. and Volovich, I. A.: The stochastic Levy Laplacian and Yang-Mills equation on manifolds; Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4 (2001) 161–172. 15. Lévy, P.: Le¸cons d’Analyse Fonctionnelle. Gauthier-Villars, Paris, 1922. 16. Obata, N.: A characterization of the Lévy Laplacian in terms of infinite dimensional rotation groups; Nagoya Math. J. 118 (1990) 111–132. 17. Obata, N.: An analytic characterization of symbols of operators on white noise functionals; J. Math. Soc. Japan 45 (1993) 421–445. 18. Obata, N.: White Noise Calculus and Fock Space. Lect. Notes in Math. Vol. 1577, SpringerVerlag, 1994. 19. Obata, N.: Quadratic quantum white noises and Lévy Laplacian; Nonlinear Anal. 47 (2001) 2437–2448. 20. Obata, N. and Ouerdiane, H.: Heat equation associated with Lévy Laplacian; in Proceedings of the Internatational Conference on Stochastic Analysis and Applications (S. Albeverio, A. B. de Monvel and H. Ouerdiane (Eds.)), (2004) 53–68, Kluwer Academic Publishers. 21. Polishchuk, E. M.: Continual Means and Boundary Value Problems in Function Spaces. Birkh¨ auser, 1988. 22. Saitˆ o, K.: A stochastic process generated by the Lévy Laplacian; Acta Appl. Math. 63 (2000) 363–373. 23. Saitˆ o, K. and Tsoi, A. H.: The Lévy Laplacian as a self-adjoint operator; in Quantum Information (Eds. T. Hida and K. Saitˆ o), (1999) 159–171, World Scientific. 24. Saitˆ o, K. and Tsoi, A. H.: The Lévy Laplacian acting on Poisson noise functionals; Infin. Dimens. Anal. Quantum Probab. Relat. Top. 2 (1999) 503–510. 25. Zhang, Y. N.: Levy Laplacian and Brownian particles in Hilbert spaces; J. Funct. Anal. 133 (1995) 425–441. Un Cig Ji: Department of Mathematics, Research Institute of Mathematical Finance, Chungbuk National University, Cheongju 361-763, Korea E-mail address: [email protected] Habib Ouerdiane: D´ epartment de Math´ ematiques, Facult´ e de Sciences de Tunis, Universit´ e de Tunis El Manar, Campus Universitaire, Tunis, 1060 Tunisia E-mail address: [email protected] ˆ: Department of Mathematics, Meijo University, Nagoya 468-8502, Kimiaki Saito Japan E-mail address: [email protected]


MARKOV SEMIGROUPS AND GROUPS OF OPERATORS JEROME A. GOLDSTEIN, ROSA MARIA MININNI, AND SILVIA ROMANELLI Abstract. We consider different realizations of the operators Lθ,a u(x) := x2a u00 (x) + (ax2a−1 + θxa )u0 (x), θ ∈ R, a ∈ R, acting on suitable spaces of real valued continuous functions. The main results deal with the existence of Feller semigroups generated by Lθ,a and the representation Lθ,a = G2a + θGa , where Ga u = xa u0 , 0 ≤ a ≤ 1, generates a (not necessarily strongly continuous) group. Explicit formulas of the generated semigroups are also deduced.

1. Introduction Let us denote by C(J ) the space of all real valued continuous functions on an interval J, having finite limits at all endpoints not included in J, equipped with the sup-norm. We are interested in the operators Lθ,a u(x) := x2a u00 (x) + (ax2a−1 + θxa )u0 (x), where θ ∈ R, a ∈ R. In [7], using Feller classification of the boundary points we showed that for any θ ∈ R, a ∈ R, the operator Lθ,a generates a Feller semigroup on C[0, +∞]. Here, in addition to analogous generation results in different spaces of continuous functions, we obtain an explicit representation of the semigroup generated by Lθ,a for suitable a. Indeed, if 0 ≤ a ≤ 1 and Ga u := xa u0 , then the operator Lθ,a can be represented as Lθ,a u = G2a u + θGa u, where Ga generates a (not necessarily strongly continuous) group. Thus a variant of Romanov’s formula applies and the results follow. For the connections with the Black-Merton-Scholes equation see [8]. In the following, for any Banach space X, C(R, X) will denote the Banach space of all X-valued continuous functions defined in R and LK (X) the Banach algebra of all linear bounded operators on a Banach space X over K, for K = R or K = C. 2. Feller semigroups, analytic semigroups and cosine functions This section provides a brief description of the basic definitions and results about Feller semigroups, analytic semigroups and cosine functions, which form a functional analytic background for our results. 2000 Mathematics Subject Classification. Primary 47D03; Secondary 47D06, 47D07. Key words and phrases. Markov semigroups, operator semigroups and groups. 247

248

JEROME A. GOLDSTEIN, ROSA MARIA MININNI, AND SILVIA ROMANELLI

Let us consider a locally compact, separable, metric space (K, ρ) and define K∂ = K ∪ ∂, where ∂ is the point at infinity, if K is not compact. Hence K∂ is compact and ∂ is some point disjoint from K if K is compact. Define C(K∂ ) to be the space of all real-valued continuous functions on K∂ . The space C(K∂ ) is a Banach space with the maximum norm kf k = supx∈K∂ |f (x)|. Observe that for K = [0, +∞), we have C(K∂ ) = C[0, +∞]. Define the subspace C0 (K) as follows C0 (K) = {f ∈ C(K∂ ) : f (∂) = 0}.

The space C0 (K) is a closed subspace of C(K∂ ), hence it is a Banach space. Note that C0 (K) can be identified with C(K) if K is compact. Let us recall the notions of C0 -semigroup and of Feller semigroup. Definition 2.1. A family (T (t))t≥0 , T (t) ∈ LK (X) is called a C0 -semigroup on X if it satisfies the following conditions: (i): T (t + s) = T (t) T (s), t, s ≥ 0; T (0) = I; (ii): (T (t))t≥0 is strongly continuous in t, i.e. lim kT (t + s) f − T (t) f k = 0, s↓0

f ∈ X, t ≥ 0.

A family (T (t))t≥0 , T (t) ∈ LR (C(K∂ )) is a Feller semigroup on C(K∂ ) if it satisfies (i) and (ii), and, in addition, the following property: (f ∈ C(K∂ ), 0 ≤ f ≤ 1 on K∂ ) ⇒

(0 ≤ T (t) f ≤ 1, t ≥ 0, on K∂ , T (t)1 = 1, t ≥ 0).

Feller semigroups can be related to particular classes of Markov transition functions, i.e. the so-called uniformly stochastically continuous transition functions, defined as follows. Definition 2.2. A transition probability function Pt , t ≥ 0, on K is said to be uniformly stochastically continuous on K if the following condition is satisfied: For each ε > 0 and each compact E ⊂ K, we have lim sup [1 − Pt (x, Uε (x))] = 0, t↓0 x∈E

where Uε (x) = {y ∈ K : ρ(x, y) < ε} is an ε-neighborhood of x. More precisely, the following result holds (see e.g. [13, Theorem 9.2.3]). Theorem 2.3. The following statements are equivalent: (a): (Pt )t≥0 is a uniformly stochastically continuous C0 -transition function on K and satisfies the condition (L) For each s > 0 and each compact E ⊂ K, it follows that lim sup Pt (x, E) = 0;

x→∂ 0≤t≤s

(b): The family of operators (Tt )t≥0 , defined by Z T (t) f (x) = Pt (x, dy) f (y), f ∈ C0 (K), K

is a Feller semigroup on C0 (K).

MARKOV SEMIGROUPS AND GROUPS OF OPERATORS

249

Notice that any Feller semigroup (and hence any corresponding family of uniformly stochastically continuous C0 transition functions) is uniquely associated to a suitable operator (A, D(A)) called the generator of the Feller semigroup, defined in the following way: the domain of A is the subspace D(A) given by T (t) u − u ∈ C0 (K) D(A) = u ∈ C0 (K) : there exists lim t↓0 t and, for any u ∈ D(A),

T (t) u − u . t According to the Lumer-Phillips version of the Hille-Yosida theorem (see e.g. [6, Theorem 3.3]), the simplest method to show that a closed, densely defined, linear operator (A, D(A)) on C0 (K) generates a C0 -contraction semigroup is to check that the operator (A, D(A)) is dissipative and satisfies the range condition. The notion of dissipativity relies on the duality map. Au = lim t↓0

Definition 2.4. Let X be a Banach space with dual space X 0 and < ·, · > be the pairing between X and X 0 . For every x ∈ X, we say that j(x) is the duality map of x if j(x) = {x0 ∈ X 0 : < x, x0 > = kxk2 = kx0 k2 }. Notice that Hahn-Banach theorem implies that j(x) 6= ∅ for any x ∈ X. In particular, for X = C0 (K), if f ∈ X, f 6= 0 and δt0 denotes the evaluation function at t0 , we have: {f (t0 ) · δt0 : t0 ∈ K, |f (t0 )| = kf k} ⊂ j(f ).

Definition 2.5. An operator (A, D(A)) on a Banach space X is called dissipative if, for any f ∈ D(A), there exists x0 ∈ j(f ) such that Re < Af, x0 > ≤ 0.

Now the Lumer-Phillips version of the Hille-Yosida theorem reads as follows (see [6, Chapter I, Section 3]). Theorem 2.6. Let (A, D(A)) be a linear operator on a Banach space X. Then (A, D(A)) generates a C0 -contraction semigroup if and only if A is densely defined and m-dissipative ( i.e. (A, D(A)) is dissipative and ρ(A) ∩ (0, +∞) 6= ∅).

Definition 2.7. A C0 -group T on a Banach space X over K is a family (T (t))t∈R of elements of LK (X) satisfying the conditions of Definition 2.1 but with R+ replaced by R. The generator (A, D(A)) of a C0 -group on X is defined by Af = lim

t→0

T (t) f − f , t

the domain of A being the subspace T (t) f − f D(A) = f ∈ X : lim ∈X . t→0 t

Remark 2.8. Note that here the limit is a two-sided one. Moreover, (A, D(A)) is the generator of a C0 -group (T (t))t∈R if and only if ±A generates a C0 -semigroup (T± (t))t≥0 , where T+ (t), t≥0 T (t) = T− (−t), t ≤ 0.

250


Observe that, for X = C0 (K), the automorphism groups on X can be characterized as follows (see e.g. [11, Propositions 3.8, 3.9]). Proposition 2.9. Let Φ : R × K −→ K be a flow (Φt )t∈R on K (i.e. Φt : K → K, Φt (x) = Φ(t, x), is continuous for any t ∈ R, and Φ0 (x) = x, x ∈ K, Φs ◦ Φt = Φs+t , s, t ∈ R). Let (ht )t∈R be a cocycle of Φ (i.e. (ht )t∈R is a family of real-valued bounded continuous functions on K such that h0 = 1 and ht+s = ht · (hs ◦ Φt ), s, t ∈ R). If, for every x ∈ K, the mappings t 7→ Φt (x),

t 7→ ht (x)

are continuous, then the operator T (t) f = ht · (f ◦ Φt ) defines a C0 -group. Conversely, if (T (t))t∈R is a C0 -group of positive operators on C0 (K), then there exist a continuous flow on K and a continuous cocycle (ht )t∈R of Φ such that T (t) f = ht · (f ◦ Φt ), for any f ∈ C0 (K), t ∈ R. Among all possible C0 -semigroups, the most regular class is the class of analytic semigroups, defined as follows. Definition 2.10. For α ∈ (0, π] we define the sector S(α) in the complex plane by S(α) = {r eiθ : r > 0, θ ∈ (−α, α)}.

A C0 -semigroup (T (t))t≥0 on a (complex) Banach space X is called a bounded analytic semigroup of angle α ∈ 0, π2 if (T (t))t>0 is the restriction of an analytic function T (·) : S(α) −→ LC (X)

satisfying i): T (z) T (z 0) = T (z + z 0 ), z, z 0 ∈ S(α); ii): For each α1 ∈ (0, α) the set {T (z) : z ∈ S(α1 )} is uniformly bounded and limn→∞ T (zn ) f = f for any null-sequence (zn ) in S(α1 ) and every f ∈ X. We say that A generates an analytic semigroup of angle α if for every ε > 0 with α − ε > 0 there is an ω = ω(ε) such that A − ωI generates a bounded analytic semigroup of angle α − ε. Observe that the generators of analytic semigroups can be characterized as follows (see e.g. [5]). Theorem 2.11. Let (A, D(A)) be a densely defined operator on a Banach space X and α ∈ 0, π2 . Then (A, D(A)) is the generator of an analytic semigroup of angle α if and only if there exists R > 0 such that π , |λ| ≥ R implies λ ∈ ρ(A) λ∈S α+ 2 and for every α1 ∈ (0, α) there exists a constant M ≥ 0 such that π M , λ ∈ S α1 + kR(λ, A)k ≤ , |λ| ≥ R. |λ| 2


251

In the case of second order Cauchy problems the corresponding notion of generator is given by means of the definition of generator of a cosine function (see e.g. [6, Chapter II, Section 8]). Definition 2.12. A (strongly continuous) cosine function on a Banach space X is a family C = (C(t))t∈R of linear bounded operators on X satisfying (i): C(t + s) + C(t − s) = 2C(t) C(s) (ii): C(0) = I; (iii): C(·) f ∈ C(R, X), for each f ∈ X.

t, s ∈ R;

The generator (A, D(A)) of a cosine function C is the operator A := C 00 (0), with domain D(A) := {f ∈ X : C(·) f ∈ C 2 (R, X)}. There are significant relations among generators of C0 -groups, generators of cosine functions and generators of analytic semigroups. We collect some of them in the following theorem (see e.g. [6, Chapter II, Section 8]). Theorem 2.13. (i) If (B, D(B)) is the generator of a C0 -group (T (t))t∈R on a Banach space X, then for any a ∈ R, the operator (aI + B 2 , D(B 2 )) generates a cosine function Ca on X. If a = 0, then (B 2 , D(B 2 )) generates a cosine function C0 given by C0 (t) =

[T (t) + T (−t)] , 2

t ∈ R (d0 Alembert0 s

f ormula).

(ii) Let (A, D(A)) be the generator of a cosine function C on a Banach space X. Then (A, D(A)) generates a C0 -semigroup (T (t))t≥0 given by Z ∞ y2 1 e− 4t C(s) f ds, t > 0 (Romanov 0 s f ormula) T (t) f = √ πt 0 for any f ∈ X. In addition, if X is a complex space, then (T (t))t≥0 is an analytic semigroup in the right half plane. 3. Feller semigroups and explicit representations in C0 (R+ ) If J = (r1 , r2 ) is a real interval, with −∞ ≤ r1 < r2 ≤ +∞, let A be a second order differential operator of the type Au := a(x)u00 + b(x)u0 , where a and b are real valued continuous functions on J such that a(x) > 0 for any x ∈ J. Then we can introduce the Feller classification of the boundary (see e.g. [5, Chapter VI Section 4]) . Let us denote by Z x Z x 1 b(s) ds , Q(x) := W (x) := exp − W (s) ds, a(x)W (x) x0 x0 a(s) R(x)

:= W (x)

Z

x x0

1 ds, a(s)W (s)

252


where x ∈ J and x0 is fixed in J. The boundary point r2 is said to be regular

if

exit

if

entrance

if

natural

if

Q ∈ L1 (x0 , r2 ),

R ∈ L1 (x0 , r2 );

Q∈ / L1 (x0 , r2 ),

R ∈ L1 (x0 , r2 );

Q ∈ L1 (x0 , r2 ),

R∈ / L1 (x0 , r2 );

Q∈ / L1 (x0 , r2 ),

R∈ / L1 (x0 , r2 ).

Analogous definitions can be given for r1 by considering the interval (r1 , x0 ) instead of (x0 , r2 ). Previous classification of the endpoints allows us to state Feller’s theorem, which characterizes when the operator A with the so-called Wentzell boundary conditions (i.e. lim Au(x) = 0) generates a Feller semigroup (see e.g. [5, Chapter VI x→r1 ,x→r2

Theorems 4.14, 4.17]), as follows. Proposition 3.1. The operator A with domain DM (A) := {u ∈ C(J ) ∩ C 2 (J) : Au ∈ C(J)} generates a Feller semigroup on C(J) if and only if r1 and r2 are of entrance or natural type. The operator A with domain D(A) := {u ∈ C(J ) ∩ C 2 (J) :

lim

x→r1 ,x→r2

Au(x) = 0}

generates a Feller semigroup on C(J ) if and only if both the endpoints r 1 and r2 are not of entrance type. If we are working in the space C[0, +∞], then the endpoints 0 and +∞ are not of entrance type for the operator Lθ,a defined by Lθ,a u(x) := x2a u00 (x) + (ax2a−1 + θxa ) u0 (x), acting on C[0, +∞] and so the following theorem holds ([7, Theorem 2]). Theorem 3.2. For any θ ∈ R, a ∈ R the operator Lθ,a with domain D(Lθ,a ) = {u ∈ C[0, +∞] ∩ C 2 (0, +∞) :

lim

x→0+ ,x→+∞

Lθ,au(x) = 0}

generates a Feller semigroup on C[0, +∞]. Similar arguments work as well even if we replace C(R+ ) by C(R− ) and the e θ,a u = (−x)2a u00 + (−a(−x)2a−1 + θ (−x)a ) u0 operator Lθ,a by the operator L having domain e θ,a ) = {u ∈ C(R− ) ∩ C 2 (−∞, 0) : D(L

lim

e θ,a u(x) = 0}. L

lim

e θ,a u(x) = 0} L

x→−∞,x→0−

Indeed, we prove the following result.

e θ,a with domain Theorem 3.3. The operator L e θ,a ) = {u ∈ C(R− ) ∩ C 2 (−∞, 0) : D(L

x→−∞,x→0−

generates a positive contraction semigroup in C(R − ).


253

Proof. In order to study the boundary −∞, let us take x0 = −1. Let us consider the cases (i) a = 1, (ii) a < 1, and (iii) a > 1. (i) For x < 0 and θ ∈ R, let us evaluate Z x Z x t + θ(−t) θ−1 Wθ (x) = exp − dt = exp − dt (−t)2 −1 −1 (−t) = exp (θ − 1) log(−t)|x−1 = (−x)θ−1 .

Consequently, for x < 0 and θ ∈ R we obtain Z x Z x 1 1 θ−1 Qθ (x) = (−t) dt and R (x) = (−t)−θ−1 dt. θ (−x)1+θ −1 (−x)1−θ −1

In particular, for θ = 0 we have

log(−x) = R0 (x). x Therefore Q0 ∈ / L1 (−∞, −1), R0 ∈ / L1 (−∞, −1), and hence −∞ is natural. For x < 0 and θ 6= 0 we obtain that Q0 (x) =

1 − (−x)θ = R−θ (x). θ(−x)1+θ Thus −∞ is natural for any θ ∈ R. Qθ (x) =

(ii) For x < 0 let us evaluate

Z Wθ (x) = exp −

x

−1

Z = exp −

x

−1

(−a)(−t)2a−1 + θ (−t)a dt (−t)2a

θ −a + (−t) (−t)a

(−t)1−a = exp − a log (−t) + θ 1−a

dt

x

−1

(−x)1−a θ = exp − a log (−x) + θ − 1−a 1−a θ

e− 1−a (1−(−x) = (−x)a

1−a

)

.

1 , (−x)a x 1 1 (−s)1−a (−x)1−a 1 = , ds = − 1− (−s)a (−x)a 1 − a −1 (−x)a 1−a

In particular, for θ = 0, W0 (x) = (−x)a Q0 (x) = (−x)2a and

Z

R0 (x) =

x −1

1 (−x)a

Z

x −1

(−s)a 1 ds = (−s)2a (−x)a

Z

x −1

1 ds = Q0 (x). (−s)a

254


Thus Q0 ∈ / L1 (−∞, −1), R0 ∈ / L1 (−∞, −1), and hence −∞ is natural. For θ 6= 0 we have 1−a Z θ (−x)a e 1−a (1−(−x) ) x Wθ (s) ds Qθ (x) = (−x)2a −1 θ

=

e 1−a (1−(−x) (−x)a

"

1−a

−θ

Z

θ

x −1

h

θ

)

ds 1−a

θ

e 1−a (−s) (−s)a

1−a

e 1−a (−x)

1−a

ds

#x

−1

1−a

θ

− e 1−a

i

h θ i 1−a 1 ) 1−a (1−(−x) e − 1 , θ (−x)a

e− 1−a (1−(−x) (−x)a θ

e− 1−a (1−(−x) = (−x)a θ

=

−1

e− 1−a (1−(−s) (−s)a

θ

e− 1−a (−x) θ (−x)a

θ

Rθ (x) =

θ

x

e 1−a (−s) − θ

1−a

θ

=−

and

Z

e 1−a (1−(−x) ) e− 1−a (−x)a

e 1−a (−x) = (−x)a

=

)

1−a

θ

=

1−a

1−a

e 1−a (−x) (−x)a

1−a

1−a

Z

x −1

)

)

Z Z

x −1 x −1 θ

θ

(−s)a e 1−a (1−(−s) (−s)2a θ

e 1−a (1−(−s) (−s)a

e− 1−a (−s) (−s)a

1−a

1−a

)

ds

)

ds

1−a

ds = Q−θ (x).

Hence Qθ ∈ / L1 (−∞, −1), Rθ ∈ / L1 (−∞, −1), and −∞ is natural.

(iii) Similar calculations as in the case (ii) yield that for any θ ∈ R we have Rθ (x) = Q−θ (x), and Qθ ∈ L1 (−∞, −1), Rθ ∈ L1 (−∞, −1). We conclude that −∞ is regular. Then, in any case, −∞ and 0 are not of entrance type and the assertion holds.

Now we focus on the operator Lθ,1 (i.e. a = 1) acting on the closed subspace C0 (R+ ). Let us define the mapping Φ : R × R+ −→ R+ , such that for any t ∈ R, x ∈ R+ Φ(t, x) = Φt (x) = x et .

It is straightforward to show that Φ0 (x) = x, Φt ◦ Φs = Φt+s , x ∈ R+ , t, s ∈ R, and, for any x ∈ R+ , the mapping t 7−→ Φt (x) is continuous.


255

According to [11, B-II, Propositions 3.8, 3.13], the operators S(t)f := f ◦ Φ t , t ∈ R, define positive bounded operators on C0 (R+ ) and (S(t))t∈R is a positive (C0 ) automorphism group on C0 (R+ ). Its generator is the closure of the operator A∞ u(x) := xu0 (x), x ∈ R+ with domain D(A∞ ) := Cc1 [0, +∞). Here Cc [0, +∞) := {f ∈ C(R+ ) : f vanishes in a neighborhood of +∞},

Cck [0, +∞) := {f ∈ C k (R+ ) : f vanishes in a neighborhood of +∞}, x ∈ R+ , f ∈ C(R+ ).

M f (x) := xf (x),

Observe that the domain of A∞ is D(A∞ ) := u ∈ C 1 (R+ ) ∩ C0 (R+ ) : M u0 ∈ C0 (R+ )

and G := A∞ generates a (C0 ) group of isometries on C0 (R+ ). Indeed for any t ∈ R, kS(t)uk∞ = sup |u(xet )| = sup |u(s)| = kuk∞ . x≥0

s≥0

2 Let us consider the square of the operator A∞ , say A∞ , given by 2 A∞ u(x) = x(xu0 )0 2

2 with domain D(A∞ ) := Cc2 [0, +∞) and the square of A∞ , say A∞ , whose domain is 2

D(A∞ ) := 2

u ∈ C 2 (R+ ) ∩ C0 (R+ ) : M u0 , M (M u0 )0 ∈ C0 (R+ ) .

2

2 . In addition, according to Theorem 2.13, G 2 = A It is clear that A∞ = A∞ ∞ generates a cosine function and the analytic semigroup (T (t)) generated by G 2 has the following Romanov representation:

1 T (t)u(x) = √ πt =

Z

1 √ 2 πt

∞

2

e

− y4t

0

Z

∞ 0

S(y)u(x) + S(−y)u(x) 2

y2 e− 4t u(x ey ) + u(x e−y ) dy,

dy

for any t with Re (t) > 0, and u ∈ C0 (R+ ), x ∈ R+ . Then (T (t))t≥0 is a C0 semigroup of contractions, since for t > 0, Z ∞ 2 1 − y4t y −y √ e u(x e ) + u(x e ) dy |T (t)u(x)| = 2 πt 0

≤

This implies

1 √ πt

Z

∞

y2

e− 4t dy

0

kuk∞ = kuk∞ .

kT (t)uk∞ = sup |T (t)u(x)| ≤ kuk∞. x≥0

Therefore, we have proved the following result

256


Theorem 3.4. The closure of the operator (A∞ , D(A∞ )) defined by A∞ u(x) = xu0 , with domain D(A∞ ) := Cc1 [0, +∞), generates 2 a positive (C0 ) group of isometries 2 on C0 (R+ ). Hence the square A∞ , D A∞ , with domain 2 D(A∞ ) := u ∈ C 2 (R+ ) ∩ C0 (R+ ) : M u0 , M (M u0 )0 ∈ C0 (R+ ) ,

generates a cosine function, and an analytic semigroup (T (t))Re(t)>0 of contractions having the following representation: Z +∞ y2 1 T (t)u(x) = √ e− 4t u(x ey ) + u(x e−y ) dy. 2 πt 0

In order to consider the operator Lθ,1 , we shall examine additional properties of 2 the operator A∞ with respect to its square A∞ . Let us remark that the operator A∞ is dissipative on C0 (R+ ). This follows from Theorem 2.6 but we give a separate direct proof. Indeed, let u ∈ D(A∞ ) and choose x0 ∈ [0, ∞) such that u(x0 ) = eiθ kuk∞ , for some real θ. If x0 > 0, then u0 (x0 ) = 0 and this implies A∞ u(x0 ) = 0. Hence < A∞ u, δx0 > = 0 and since δx0 e−iθ kuk∞ ∈ j(u), we are done. In the case x0 = 0, u(0) = kuk∞, and u is real valued, then for any x > 0 u(x) ≤ u(0). Thus lim supx→0 u0 (x) ≤ 0 and so A∞ u(0) ≤ 0. Therefore, we can conclude that A∞ is dissipative, in case u is real valued. The proof for u complex is a trivial variant and we omit it. Notice that 2

2 + (1 + θ) A Lθ,1 = A∞ ∞ = A∞ + (1 + θ) A∞ .

Similar arguments as in [5, Chapter III, Example 2.2] imply that (1 + θ) A ∞ is 2 2 A∞ - bounded with A∞ - bound equal to 0. Moreover [5, Chapter III, Lemma 2.4] 2 2 yields that A∞ + (1+θ) A∞ with domain D A∞ is closed and, as a consequence of [5, Chapter III, Theorem 2.7], generates a contraction semigroup on C0 (R+ ), which is analytic in the right half plane. In addition, the semigroup generated by 2 A∞ + (1 + θ) A∞ is given by U (t) = T (t) S((1 + θ) t),

t ≥ 0.

Hence, if u ∈ C0 (R+ ) the semigroup has the explicit representation Z ∞ y2 1 √ U (t) u(x) = e− 4t [S(y) S((1 + θ)t) u(x) + S(−y) S((1 + θ) t) u(x)] dy 2 πt 0 Z ∞ y2 1 = √ e− 4t [S(y + (1 + θ) t) u(x) + S(−y + (1 + θ) t) u(x)] dy 2 πt 0 Z ∞ y2 1 = √ e− 4t [u x ey+t(1+θ) + u x e−y+t(1+θ) ] dy. 2 πt 0 This is valid for all t > 0 and x ∈ R+ . Therefore the following result holds.


257

2 Theorem 3.5. The closure of the operator Lθ,1 with domain D A∞ generates a positive contraction analytic semigroup (U (t))t∈R+ on C0 (R+ ) having the following explicit representation: Z ∞ y2 1 U (t) u(x) = √ e− 4t [u x ey+t(1+θ) + u x e−y+t(1+θ) ] dy, 2 πt 0 for all t with Re t > 0, and all x ∈ R+ .

4. Explicit representations in C(R) Let us consider the Banach space C(R), equipped with the sup-norm, and the operator Lθ,a defined in the Introduction. For a = 0, the operator Lθ,0 is given by Lθ,0 u(x) = u00 (x) + θu0 (x). It is well known that the operator Gu(x) := u0 (x) with domain D(G) := {u ∈ C(R) : u0 ∈ C(R)} generates the translation group (S(t))t∈R on C(R), where S(t)u(x) := u(x + t), x, t ∈ R. Hence, according to Theorem 2.13, the square G2 with domain D(G2 ) = {u ∈ C(R) : u0 , u00 ∈ C(R)}

generates a cosine function and the (analytic) semigroup generated by G2 has the following representation Z ∞ y2 1 T (t)u(x) = √ e− 4t [u(x + y) + u(x − y)] dy 2 πt 0 for any t > 0, u ∈ C(R), and x ∈ R. In addition, according to [6], any u ∈ D(G2 ) satisfies lim u0 (x) = 0 = lim u00 (x). x→±∞

x→±∞

It follows that, for any θ ∈ R, the semigroup (U (t))t≥0 generated by Lθ,0 = G2 + θG in C(R) can be written as T (t) S(θt) and we have Z ∞ y2 1 U (t)u(x) = √ e− 4t [u(x + θt + y) + u(x + θt − y)] dy 2 πt 0 for any t > 0, u ∈ C(R) and x ∈ R. Our next aim is to show that our operator Lθ,a can be interpreted as an operator of the type G2a +θGa , where Ga in some sense generates a suitable group on C(R). Definition 4.1. Let x be a real number and a be a positive number, then we define ( xa if x ≥ 0; {a} x = − (−x)a if x < 0. Observe that x{1} = x for any real number x.

258


Lemma 4.2. For any real number x and any a > 0 and b > 0, we have {b} x{a} = x{ab} ,

and

(−x) Moreover, for any x 6= 0

{a}

= −x{a} .

d {a} x = a |x|a−1 . dx Proof. Concerning the first assertion, if x ≥ 0, it is trivial. If x < 0, then we have x{a} < 0, and thus {b} x{a} = (−(−x)a ){b} = −((−x)a )b = −(−x)ab = x{ab} .

The second and third part of the assertion easily follow from the definition of x{a} . Now we are in position to prove Theorem 4.3. Let us assume that 0 < a < 1 and for any h ∈ C(R), x ∈ R, define for any t ∈ R: h 1 i{ 1−a } {1−a} . T (t) h(x) = h x + (1 − a) t

Then (T (t))t∈R is a positive contraction group of operators on C(R). In addition, for any h ∈ C(R) and x ∈ R, the mapping t → T (t)h(x) is continuous, and for d any h ∈ C(R) ∩ C 1 (R) and x ∈ R there exists (T (t)h(x))|t=0 = |x|a h0 (x). dt

Proof. First observe that for any t ∈ R, T (t) is a linear bounded operator on C(R), which is positive and contractive. It is also clear that for any h ∈ C(R) and x ∈ R we have 1 { 1−a } {1−a} T (0) h(x) = h x = h(x) by virtue of Lemma 4.2. This yields that T (0) = I. Let us proceed to show that for any t, s ∈ R, h ∈ C(R) T (t + s) h(x) = T (t) T (s) h(x), Indeed, T (t) T (s) h(x) = T (t) h =h

=h

h h

h

x{1−a} + (1 − a) s

x{1−a} + (1 − a) s

x ∈ R.

1 i{ 1−a }

1 i{ 1−a } {1−a}

+ (1 − a) t

1 i { 1−a } x{1−a} + (1 − a) s + (1 − a) t

(4.1)

1 !{ 1−a }


=h

h

x{1−a} + (1 − a) (t + s)

= T (t + s) h(x).

259

1 i{ 1−a }

Thus (4.1) is proved. Now let us observe that for any h ∈ C(R) and x ∈ R h 1 i{ 1−a } {1−a} T (t)h(x) − h(x) = h x + (1 − a) t − h(x).

It follows that lim T (t)h(x) − h(x) = 0. This gives the continuity of the mapping t→0

t → T (t)h(x) at t = 0. In addition, there easily follows the continuity at any t ∈ R. Finally, if h ∈ C(R) ∩ C 1 (R), x ∈ R and t ∈ R, t 6= 0, let us examine T (t)h(x) − h(x) . We have t 1 { 1−a } {1−a} − h(x) h x + (1 − a) t T (t)h(x) − h(x) = . t t Then, taking the limit as t → 0, an easy consequence of de l’Hospital rule gives T (t)h(x) − h(x) = h0 (x) |x|a and the proof has been completed. lim t→0 t Corollary 4.4. Let us assume that 0 < a < 1 and that (T (t))t∈R is the group of operators on C(R) defined in Theorem 4.3 and let us denote by Ce (R) := {h ∈ C(R) : h is even}. Then for any h ∈ Ce (R), we have T (t) h(−x) = T (−t) h(x),

t ∈ R, x ∈ R.

(4.2)

Proof. Let us fix h ∈ C(R), h even (i.e. h(−x) = h(x), for any x ∈ R). In order to prove (4.2) we observe that h 1 i{ 1−a } T (t) h(−x) = h (−x){1−a} + (1 − a) t =h

=h

h h

−x x

{1−a}

{1−a}

+ (1 − a) t

− (1 − a) t

1 i{ 1−a }

1 i{ 1−a }

= T (−t) h(x). Corollary 4.5. For any f ∈ C(R+ ) let us denote by fe the even extension of f to R. Let a ∈ (0, 1) and (T (t))t∈R be the group of operators on C(R) defined in Theorem 4.3. Then the family of operators (Te(t))t∈R defined as follows e Te(t) f (x) = T (t) f(x),

t ∈ R, x ∈ R+ ,

260


is a positive contraction group on C(R+ ). In addition, for any f ∈ C(R+ ) and x ∈ R+ the mapping t → T (t)f (x) is continuous, and for any f ∈ C(R + ) ∩ C 1 (R+ ) d and x ∈ R+ there exists T (t)f (x) |t=0 = xa f 0 (x). dt Proof. From Corollary 4.4, for any f ∈ C(R+ ) we deduce that ( T (t) f (x), t ≥ 0, x ≥ 0; e T (t) f (x) = e T (−t) f(−x), t < 0, x ≥ 0.

Hence, by taking into account Theorem 4.3 and the previous Corollary, the assertion holds. Note that the previous groups (T (t))t∈R (respectively, (Te(t))t∈R ) on C(R) (respectively, on C(R+ )), have some regularity properties, as the continuity of the map t → T (t)f (x) (respectively, t → Te(t)f (x)), for any fixed f ∈ C(R), x ∈ R (reT (h)f (x) − f (x) spectively, f ∈ C(R+ ), x ∈ R+ ) and the pointwise convergence of h Te(h)f (x) − f (x) e (x)). Also, (respectively ), as h → 0, to Af (x) (respectively to Af h (T (t))t∈R (respectively, (Te(t))t∈R ) is strongly measurable on C(R) (respectively, on C(R+ )), hence strongly continuous at t for all t 6= 0 (see [10, Theorem 10.2.3]). Finally, by using the results by Priola [12, Chapter 6], we can conclude that our groups are C0 -groups on the respective spaces, provided that we identify the elements of C(R) (respectively C(R+ )) with the elements of C(Rc ) (respectively C(Rc+ )). Here Rc (respectively Rc+ ) denotes the Alexandroff compactification of R (respectively R+ ). For the connections with integrated semigroups see also [1]. All these facts allow us to repeat similar arguments as in Section 3 in order to give an explicit representation of the semigroups on C(R) (respectively on C(R + )) generated by the operators Lθ,au(x) = x2a u00 (x) + (a|x|2a−1 + θ|x|a )u0 (x) = G2a u(x) + (1 + θ)Ga u(x),

x ∈ R,

where (Ga , D(Ga )) is defined as follows D(Ga ) = {u ∈ C(R) ∩ C 1 (R) : u0 (·)|x|a ∈ C(R)}, Ga u(x) = u0 (x)|x|a ,

(respectively,

u ∈ D(Ga ),

x∈R

e θ,au(x) = x2a u00 (x) + (a x2a−1 + θ xa )u0 (x) = G e2a u(x) + (1 + θ)G e a u(x), L e a , D(G ea )) is defined as follows where (G

ea ) = {f ∈ C(R+ ) ∩ C 1 (R+ ) : f 0 (·)xa ∈ C(R+ )}, D(G

ea f (x) = f 0 (x) xa , G

ea ), f ∈ D(G

x ∈ R+ ).

x ∈ R+ ,

More precisely, we have that the operator Lθ,a has domain D(G2a ), and, respec2 e θ,a has domain D(G fa ). Consequently, Lθ,a generates the tively, the operator L semigroup (U (t))t≥0 given by


1 U (t)u(x) = √ 2 πt

Z

∞

261

y2

e− 4t [T (y)T ((1 + θ)t)u(x) + T (−y)T ((1 + θ)t)u(x)] dy,

0

e θ,a generates the semigroup for all t > 0, u ∈ C(R) and x ∈ R. In a similar way L e (U (t))t≥0 given by e (t)u(x) = √1 U 2 πt

Z

∞

0

2

y e− 4t [Te(y)Te((1 + θ)t)u(x) + Te(−y)Te((1 + θ)t)u(x)] dy,

for all t > 0, u ∈ C((R+ ) and x ∈ R+ . e (t))t≥0 . Indeed, for This gives an explicit representation of (U (t))t≥0 and (U any y ≥ 0, t > 0, u ∈ C(R), x > 0 such that x1−a + (1 − a)(1 + θ)t ≥ (1 − a)y we define I1 (x, y) := T (y) T ((1 + θ)t)u(x) + T (−y) T ((1 + θ)t)u(x) 1

1

= T (y) u([x1−a + (1 − a)(1 + θ)t] 1−a ) + T (−y) u([x1−a + (1 − a)(1 + θ)t] 1−a ) 1

1

= u([x1−a + (1 − a)[(1 + θ)t + y]] 1−a ) + u([x1−a + (1 − a)[(1 + θ)t − y]] 1−a ). Analogously, for x > 0 with x1−a + (1 − a)(1 + θ)t < (1 − a)y we have I2 (x, y) := T (y) T ((1 + θ)t)u(x) + T (−y) T ((1 + θ)t)u(x) 1

1

= u([x1−a + (1 − a)[(1 + θ)t + y]] 1−a ) + u(−[−x1−a − (1 − a)[(1 + θ)t − y]] 1−a ). Then, for any t > 0, u ∈ C(R) and x > 0 it yields that Z z Z ∞ 2 y2 1 − y4t U (t)u(x) = √ e I1 (x, y) dy + e− 4t I2 (x, y) dy, 2 πt 0 z x1−a + (1 − a)(1 + θ)t . On the other hand, for x < 0 with |x|1−a ≤ 1−a (1 − a)(1 + θ)t, we have Z w Z ∞ 2 y2 1 − y4t e I3 (x, y) dy + e− 4t I4 (x, y) dy, U (t)u(x) = √ 2 πt 0 w where z :=

where w :=

−|x|1−a +(1−a)(1+θ)t , 1−a

and

I3 (x, y) := 1

1

1

1

u([−|x|1−a + (1 − a)[(1 + θ)t + y]] 1−a ) + u([−|x|1−a + (1 − a)[(1 + θ)t − y]] 1−a ),

I4 (x, y) :=

u([−|x|1−a + (1 − a)[(1 + θ)t + y]] 1−a ) + u(−[|x|1−a − (1 − a)[(1 + θ)t − y]] 1−a ).

In a similar way one can describe explicitly the case x < 0 with |x|1−a > (1 − a)(1 + θ)t. In analogy, by taking into account Corollary 4.5, one can describe e U(t)u(x) for any t > 0, u ∈ C(R+ ), x ≥ 0.

262


In [9] we apply the results obtained in this paper to some problems arising in financial mathematics. Acknowledgement. We thank the referee for helpful comments which shortened the proof of Theorem 4.3. References [1] Arendt, W.: Resolvent positive operators, Proc. London Math. Soc. 54(3) (1997), 321–349. [2] Clément, Ph. & Timmermanns, C.A.: On C0 -semigroups generated by differential operators satisfying Ventcel’ boundary conditions, Indag. Math. 89 (1986), 379–387. [3] Black, F. & Scholes, M.: The pricing of options and corporate liabilities, J. Polit. Econom. 81 (1973), 637–659. [4] Dynkin, E.E.: Markov Processes Vols. 1-2, Die Grundlehren der Math. Wissenschaften 121122, Springer-Verlag, Berlin-G¨ ottingen-Heidelberg, 1965. [5] Engel, K.J. & Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics 194, Springer, 2000. [6] Goldstein, J.A.: Semigroups of Linear Operators and Applications, Oxford Mathematical Monographs, Oxford, 1985. [7] Goldstein, J.A., Mininni, R.M. & Romanelli, S.: Generators of Feller semigroups with coefficients depending on parameters and optimal estimators, Discrete and Continuous Dynamical Systems 8 (2) (2007) (to appear). [8] Goldstein, J.A., Mininni, R.M. & Romanelli, S.: A new explicit formula for the solution of the Black-Merton-Scholes equation (2006) (Submitted). [9] Goldstein, J.A., Mininni, R.M. & Romanelli, S.: Markov semigroups and estimating functions, with applications to some financial models, Communications on Stochastic Analysis (2007) (to appear). [10] Hille, E. & Phillips, R.S.: Functional Analysis and Semi-Groups, Amer. Math. Soc. Colloquium Publications 31, 1957. [11] Nagel, R. (Ed.): One-Parameter Semigroups of Positive Operators, Lect. Notes in Mathematics 1184, Springer, 1986. [12] Priola, E.: Partial differential equations with infinitely many variables, Tesi di Dottorato, Universit` a degli Studi di Milano, 1999. [13] Taira, K.: Diffusion Processes and Partial Differential Equations, Academic Press, San Diego, 1988. Jerome A. Goldstein: Department of Mathematical Sciences, The University of Memphis, 373 Dunn Hall, Memphis TN 38152-3240, USA E-mail address: [email protected] URL: http://www.msci.memphis.edu/faculty/goldsteinj.html ` di Bari, Via Orabona Rosa Maria Mininni: Dipartimento di Matematica, Universita 4, 70125 Bari, Italy E-mail address: [email protected] ` di Bari, Via Orabona 4, Silvia Romanelli: Dipartimento di Matematica, Universita 70125 Bari, Italy E-mail address: [email protected] URL: http://www.dm.uniba.it/∼romanelli


CONVEX COMPARISON INEQUALITIES FOR EXPONENTIAL JUMP-DIFFUSION PROCESSES JEAN-CHRISTOPHE BRETON AND NICOLAS PRIVAULT Abstract. Given (Mt )t∈R+ and (Mt∗ )t∈R+ respectively a forward and a backward exponential martingale with jumps and a continuous part, we prove that E[φ(Mt Mt∗ )] is non-increasing in t when φ is a convex function, provided the local characteristics of the stochastic logarithms of (M t )t∈R+ and of (Mt∗ )t∈R+ satisfy some comparison inequalities. As an application, we deduce bounds on option prices in markets with jumps, in which the underlying processes need not be Markovian. In this setting the classical propagation of convexity assumption for Markov semigroups [4] is not needed.

1. Introduction Bounds on option prices with convex payoff functions have been obtained by several authors. Theorem 6.2 of [4], for example, states that E[φ(ST ) | S0 = x] ≤ E[φ(ST∗ ) | S0∗ = x],

x > 0,

(1.1)

for any convex function φ, provided S and S ∗ are price processes of the form dSt = rt dt + σt dWt St and

where (Wt )t∈R+

dSt∗ = rt dt + σ ∗ (t, St∗ )dWt , St∗ is a standard Brownian motion, under the condition |σt | ≤ |σ ∗ (t, St )|,

t ∈ R+ .

The proof of (1.1) relies on the backward Kolmogorov equation, provided the Markov semigroup of (St∗ )t∈R+ propagates convexity. A first extension of this type of bound to the jump-diffusion case can be found in [1], and more general results have been later proved in [2] under refined conditions, still under the propagation of convexity hypothesis. Note however that the propagation of convexity property is not always satisfied, even in the (Markovian) jump-diffusion case, cf. e.g. Theorem 4.4 in [3]. In this paper we prove a convex comparison inequality of the form E[φ(Mt Mt∗ )] ≤ E[φ(Ms Ms∗ )],

0 ≤ s ≤ t,

(1.2)

2000 Mathematics Subject Classification. Primary 60E15; Secondary 60H05, ,60G44, 60G55. Key words and phrases. Convex concentration, forward-backward stochastic calculus, exponential jump diffusion processes, option prices. 263

264

J-C BRETON AND N. PRIVAULT

where Mt , Mt∗ are respectively forward and backward exponential martingales with jumps and continuous parts, satisfying some conditions. More precisely, (1.2) will hold for convex function φ : R → R, provided the local characteristics of the stochastic logarithms of (Mt )t∈R+ and of (Mt∗ )t∈R+ satisfy the comparison inequalities assumed in Theorem 3.1 below. Our proof relies on arguments of [6], with the difference that we consider products instead of sums of forward and backward martingales. Moreover the results of [6] require a.s. uniform bounds on the diffusion coefficients which cannot be satisfied for stochastic exponentials. In some results we assume in addition that φ0 is convex, a condition that can be realized in applications when φ is e.g. an exponential payoff function. If further E[Mt∗ |FtM ] = 1, t ∈ R+ , where (FtM )t∈R+ denotes the filtration generated by (Mt )t∈R+ , then Jensen’s inequality yields E [φ(Mt )] = E φ Mt E[Mt∗ |FtM ] = E φ E[Mt Mt∗ |FtM ] ≤ E E φ(Mt Mt∗ )|FtM = E [φ(Mt Mt∗ )] ≤ E [φ (Ms Ms∗ )] ,

0 ≤ s ≤ t,

and in particular E [φ(Mt )] ≤ E [φ (M0 M0∗ )] ,

t ≥ 0.

(1.3)

We prove (1.2) using forward-backward stochastic calculus, assuming only the convexity of φ, and without propagation of convexity, cf. Theorem 3.1. We note that (1.3) can be read as a bound on option prices, where φ is a convex payoff function and Mt is the price of an underlying asset. More precisely, cf. Corollaries 4.2, 5.1 and 6.2, it yields bounds of the form E[φ(ST ) | S0 = x] ≤ E[φ(ST∗ ) | S0∗ = x], where (St )t∈R+ and

(St∗ )t∈R+

x > 0,

(1.4)

are jump-diffusion price processes of the form

dSt = rt dt + σt dWt + Jt− (dZt − λt dt), S t− where (Wt )t∈R+ is a standard Brownian motion, (Zt )t∈R+ is a point process of (stochastic) intensity λt , and (St∗ )t∈R+ can be taken as the solution of dSt∗ ˆ t + J ∗− (dN ˆt − λ∗ dt), = rt dt + σt∗ dW t t St∗− ˆ t )t∈R+ is a standard Brownian motion and (N ˆt )t∈R+ is a Poisson process where (W ∗ of (deterministic) intensity λt , mutually independent and independent of (Wt )t∈R+ and of (Nt )t∈R+ , provided the three conditions |σt | ≤ |σt∗ |,

0 ≤ Jt ≤ Jt∗ ,

Jt λt ≤ Jt∗ λ∗t ,

t ∈ [0, T ],

(1.5)

ˆt )t∈R+ to drive the are satisfied. The choice of the standard Poisson process (N ∗ jump part of (St )t∈R+ is made here to simplify the formulation of the hypotheses in (1.5). More general point processes can be actually considered, cf. Sections 4, 5 and 6.

COMPARISON INEQUALITIES FOR JUMP-DIFFUSION

265

Here, the coefficients rt , σt , Jt , σt∗ , Jt∗ , are (random) F M -adapted processes and need not be diffusion coefficients. The difference between (1.4) and (1.1) is ˆ t )t∈R+ and (N ˆt )t∈R+ are independent of that in (1.4) the integrator processes (W (σt∗ )t∈R+ and of (Jt∗ )t∈R+ . Denoting by BS(φ, x, t, r ∗ , σ ∗ , J ∗ ) the conditional Black-Scholes price BS(φ, x, t, r, σ ∗ , J ∗ ) = E[φ(St∗ )| W, Z, S0∗ = x], (1.4) reads E[φ(St ) | S0 = x] ≤ E[BS(φ, x, t, r∗ , σ ∗ , J ∗ )] between E[φ(St ) | S0 = x] and the averaged Black-Scholes price E[BS(φ, x, t, r∗ , σ ∗ , J ∗ )]. In the diffusion case when J ∗ = λ∗ = 0 and σt∗ is deterministic, our result coincides with those of the above mentioned papers, and in particular with (1.1) or Theorem 6.2 of [4]. In the jump-diffusion case, still taking σt∗ , Jt∗ , λ∗t deterministic, we get E[φ(ST ) | S0 = x] ≤ BS(φ, x, T, r∗ , σ ∗ , J ∗ ), but our hypothesis differ from those of [2] where convex ordering of the jump caracteristics is required, see Theorem 2.3 therein, whereas here our conditions are directly formulated in terms of Jt , λt , Jt∗ and λ∗t . In the general case where σt∗ , Jt∗ , λ∗t are random, our results can not be compared since our process (St∗ )t∈R+ is no longer a diffusion process as in [2]. We proceed as follows. In Section 2 we recall the framework of [6] on forwardbackward stochastic calculus. In Section 3 we prove our convex concentration inequalities for exponential martingales following the arguments of [6], in which sums are replaced by products. Applications to point processes and Poisson random measures in view of option pricing are given in Sections 4, 5, 6. 2. Forward-backward stochastic calculus In this section we recall some definitions and results on forward-backward stochastic calculus, see [6] for details. Let (Ω, F, P ) be a probability space equipped with an increasing filtration (Ft )t∈R+ and a decreasing filtration (Ft∗ )t∈R+ . Consider (Xt )t∈R+ an Ft -forward martingale with X0 = 0, and (Xt∗ )t∈R+ an Ft∗ ∗ backward martingale with X∞ = 0, such that (Xt )t∈R+ has right-continuous paths ∗ with left limits and (Xt )t∈R+ has left-continuous paths with right limits. Denote respectively by (Xtc )t∈R+ and (Xt∗c )t∈R+ the continuous parts of (Xt )t∈R+ and of (Xt∗ )t∈R+ , and by ∆Xt = Xt − Xt− ,

∆∗ Xt∗ = Xt∗ − Xt∗+ ,

their forward and backward jumps. Denote by X µ(dt, dx) = 1{∆Xs 6=0} δ(s,∆Xs ) (dt, dx), s>0

and µ∗ (dt, dx) =

X s>0

1{∆∗ Xs∗ 6=0} δ(s,∆∗ Xs∗ ) (dt, dx),

266

J-C BRETON AND N. PRIVAULT

the jump measures of (Xt )t∈R+ and (Xt∗ )t∈R+ , by ν(dt, dx) and ν ∗ (dt, dx) their respective (Ft )t∈R+ and (Ft∗ )t∈R+ -dual predictable projections, and by ([X, X])t∈R+ , ([X ∗ , X ∗ ])t∈R+ , resp. hX c , X c it , hX ∗c , X ∗c it t ∈ R+ , their optional, resp. predictable quadratic variations, which constitute the local characteristics of (X t )t∈R+ , cf. [5] in the forward case. We will use the following Itˆ o type change of variable formula for forward-backward martingales which has been proved in [6], Theorem 8.1: f (Xt , Xt∗ ) − f (X0 , X0∗ ) Z Z t 1 t 2 ∂1 f (Xu , Xu∗ )dhX c , X c iu = ∂1 f (Xu− , Xu∗ )dXu + 2 + 0 0 X + (f (Xu , Xu∗ ) − f (Xu− , Xu∗ ) − ∆Xu ∂1 f (Xu− , Xu∗ )) 0 ε(l,r) (j) = −1 : j = 1, . . . , p is similar. If instead there exists j = 2, . . . , p such that ii) is not verified, then, by the (l,r)

non crossing principle, on the right hand side of Aε(l,r) (j) gj,tk

there exists an

+

annihilator coupled with A (tk ∧ t, tk+1 ∧ t) thus giving zero. The last part is trivial.

The result above ensures that p must be even, i.e. p = 2n. From now on we introduce the notation ε(l,r) = ε(l,r) (2n) , . . . , ε(l,r) (1) ∈ {−1, 1}2n + to express that the partition ε(l,r) realizes conditions i), ii) of the Lemma above. Moreover it is 2n well known (see [7] for details) that ε(l,r) ∈ {−1, 1}+ induces a unique non crossing pair partition on the set {1, . . . , 2n} , denoted by ln(l,r) , rn(l,r) , . . . , (l1 , r1 ) , which can be assumed increasingly ordered with respect to the left indices lj ’s. As in [2] we introduce the depth function for a given partition. Definition 3.5. For any n ∈ N and ε ∈ {−1, 1}n the map dε : {1, . . . , n} → {0, ±1, . . . , ±n} defined as dε (j) :=

j X

ε (k) = |{ε (k) : ε (k) = 1, k < j}| − |{ε (k) : ε (k) = −1, k < j}|

k=1

such that for any j = 1, . . . , n, is called the depth function of ε. For any sequence of operators of the type considered above, dε (j) is the number of creators (annihilators if negative) which are on the right hand side of Aε(j) or, equivalently, the number of pairs containing j in their ”interior”. The following definition is given in order to prove a useful result for the last semi-martingale estimate. Definition 3.6. A non crossing pair partition {lj , rj }nj=1 of {1, 2, . . . , 2n} such that l1 < l2 < . . . < ln is called connected if for any k = 1, . . . , n − 1 one has n {lk , rk } ⊂ {ln , rn }. A subset {li , ri }i∈I , I ⊆ {1, . . . , n} , of {lj , rj }j=1 is called a n connected component of {lj , rj }j=1 if it is a connected non crossing pair partition.

QSC ON IFS: SEMIMARTINGALE ESTIMATES AND STOCHASTIC INTEGRAL

331

n

A non crossing pair partition {lj , rj }j=1 is called interval partition if, for any j = 1, . . . , n, lj = rj + 1. n

Lemma 3.7. Let us suppose that the pair partition {lj , rj }j=1 induced by ε ∈ 2n

{−1, 1}+ is such that 1 = rn < . . . < r1 < l1 < . . . < ln = 2n. Then, for any gl1 , . . . , gln , gr1 , . . . , grn ∈ H, any d ∈ N, ξ ∈ Hd , xd , . . . , x1 ∈ R+ one has: A (gln ) · · · A (gl1 ) A+ (gr1 ) · · · A+ (grn ) ξ (xd , . . . , x1 )   Z n λd+dε (l1 ) (yln , . . . , yl1 , xd , . . . , x1 ) Y g lj grj ylj dylj ξ  (xd , . . . , x1 ) = λd (xd , xd−1 , . . . , x1 ) Rn + j=1

where yln , . . . , yl1 ∈ R+ and dε (l1 ) = n.

Proof. The proof is straightforward by noticing that dε (lj ) − 1 = dε (lj+1 ).

As a consequence of the lemma above, any sequence of operators indexed by 2n a pair partition ε ∈ {−1, 1}+ such that dε (l1 ) = n, once applied to a d-particle vector, give only one fraction of the λn ’s, as in the case of a single pair of operators. The difference consists in the fact here the interacting functions in each fraction are no longer index consecutive. Now we investigate the case in which a sequence of annihilators and creators acting on a certain vector induces a more general connected pair partition. Lemma 3.8. Let us given a non-crossing pair partition {lj , rj }nj=1 such that for any j = 1, . . . , n−1, lj = rj +1, ln = 2n, rn = 1. Then, for any gl1 , . . . , gln , gr1 , . . . , grn ∈ H, any d ∈ N, ξ ∈ Hd , xd , . . . , x1 ∈ R+ one has: A (gln ) A gln−1 A+ grn−1 · · · A (gl1 ) A+ (gr1 ) A+ (grn ) ξ (xd , . . . , x1 ) "Z λd+2 (yln , yl1 , xd , . . . , x1 ) g ln grn (yln ) g l1 gr1 (yl1 ) × = dyl1 dyln n λ (x , . . . , x ) d d 1 R+    n−1 Y λd+2 yln , ylj , xd , . . . , x1 × glj grj ylj dylj  ξ  (xd , . . . , x1 ) λ (y d+1 l n , xd , . . . , x 1 ) j=2 where yl1 , . . . , yln ∈ R+ .

Proof. In fact A (gln ) · · · A (gl1 ) A+ (gr1 ) A+ (grn ) ξ (xd , . . . , x1 ) "Z λd+2 (yln , yl1 , xd , . . . , x1 ) g l1 gr1 (yl1 ) × = dyl1 λd+1 (yln , xd , . . . , x1 ) R+ Z λd+2 (yln , yl2 , xd , . . . , x1 ) × dyl2 gl2 gr2 (yl2 ) × λd+1 (yln , xd , . . . , x1 ) R+ Z λd+2 yln , yln−1 , xd , . . . , x1 ×···× dyln−1 g ln−1 grn−1 yln−1 × λd+1 (yln , xd , . . . , x1 ) R+ # Z λd+1 (yln , xd , . . . , x1 ) × dyln gln grn (yln ) ξ (xd , . . . , x1 ) λd (xd , . . . , x1 ) R+

332

VITONOFRIO CRISMALE

and the thesis follows.

From now on we will speak of ”fractions of the λn ’s” referred to fractions which can not be further simplified (i.e. they are irreducible); we speak of ”product of fractions of the λn ’s” referred to a product which can not be further simplified: as a consequence, we can enumerate how many factors there are in a certain product of fractions of the λn ’s. Hence, given a connected pair partition of creation-annihilation operators acting on a vector ξ, satisfying the assumptions of the lemma above, the number of fractions of the λn ’s is exactly given by the number of the index consecutive pairs. This, together with Lemma 3.7, suggests us to generalize such a result for an arbitrary sequence of annihilators and creators inducing a connected pair partition. n

Proposition 3.9. Let us given a non crossing pair partition {lj , rj }j=1 such that ln = 2n, rn = 1 and denote n o n k := {lh , rh } ⊆ {lj , rj }j=1 : rh = lh − 1, h = 1, . . . , n Then, after computing

Aε(ln ) (gln ) · · · A glj · · · A+ grj · · · Aε(rn ) (grn ) ξ

there appear exactly a product of k fractions of the λn ’s.

Proof. The thesis can be obtained by iteration. Let us fix the first pair of consecutive left-right indices from the right in the sequence, say {lk1 , rk1 } . If lk1 + 1 is a right index or rk1 − 1 is a left index, we turn to the successive index consecutive pair. On the contrary, if lk1 +1 is a left index and rk1 −1 is a right index, by the non crossing principle, on the right hand side of A+ grk1 −1 there appear only creation operators. By Lemma 3.7, the action of A glk1 +1 A glk1 A+ grk1 A+ grk1 −1 on the d + j particle vector on the right hand side (j = 1, . . . , n − 1), give rise to a unique fraction of the λn ’s. After we repeat the same arguments for all the pairs of consecutive left-right indices, finally obtaining the same type of partition described in Lemma 3.8. Let us take Fl (tk ) , Fr (tk ) ∈ Btk and introduce the following notation: ∗

2n

(Fl (tk )) Fr (tk ) = Aε(2n) (g2n ) · · · Aε(1) (g1 ) , ε ∈ {−1, 1}+

(3.9)

and denote (l,r)

N tk (l,r)

i.e. N tk

:= |{{lj , rj } , j = 1, . . . , n : rj = lj − 1}|

is the number k introduced in Proposition 3.9.

Lemma 3.10. For any d ∈ N and ξ = gd ⊗ . . . ⊗ g1 ∈ D

ξ, A (tk ∧ t, tk+1 ∧ t) (Fl (tk ))∗ Fr (tk ) A+ (tk ∧ t, tk+1 ∧ t) ξ Z tk+1 ∧t (l,r)

∗ dy (Fl (tk )) Fr (tk ) ξ ≤ M 2N tk −1 ξ, tk ∧t

(3.10)


333

Proof. Firstly we notice the sequence A (tk ∧ t, tk+1 ∧ t) (Fl (tk ))∗ Fr (tk ) A+ (tk ∧ t, tk+1 ∧ t) (l,r)

determines a connected pair partition with N tk

index consecutive pairs. Then, (l,r)

by Proposition 3.9, its action on ξ gives exactly N tk

fractions of the λn ’s. The

(l,r) N tk .

proof of (3.10) is given by induction on In fact, ε being defined in (3.9), 2(n+1) 0 take ε ∈ {−1, 1} such that  0  ε (1) = 1 ε0 (j) = ε (j − 1) , j = 2, . . . , n + 1 (3.11)  0 ε (2n + 2) = −1 n+1 (l,r) Let us suppose N tk = 1. If lj0 , rj0 j=1 is the non-crossing pair partition deter0 0 0 0 mined by ε0 , then rn+1 < rn0 < . . . < r10 < l10 < . . . < ln0 < ln+1 with ln+1 , rn+1 the indices relative respectively to A (tk ∧ t, tk+1 ∧ t) and A+ (tk ∧ t, tk+1 ∧ t) . By Lemmata 3.7 and 3.1 one has

ξ, A (tk ∧ t, tk+1 ∧ t) (Fl (tk ))∗ Fr (tk ) A+ (tk ∧ t, tk+1 ∧ t) ξ * Z tk+1 ∧t Z λd+d 0 , . . . , yl 0 y , x , . . . , x d 1 l 0 0 ε (l ) 1 n+1 1 0 dyln+1 = ξ, λ (x , . . . , x1 ) n d d tk ∧t R  +  n Y × g lj0 grj0 ylj0 dylj0  ξ j=1 * Z Z λd+d tk+1 ∧t 0,...,y 0 y [ , x , . . . , x d 1 −1 l l 0 ε0 (l ) 1 n+1 1 0 ≤ M ξ, dyln+1 × λ (x , . . . , x1 ) n d d tk ∧t R  +  n Y × g l 0 gr 0 yl 0 dyl 0  ξ j

j

j

j

j=1

Since dε0 (l 0 ) = dε(lj ) + 1 and lj0 = lj for any j = 1, . . . , n, the quantity above is j equal to Z tk+1 ∧t Z λd+dε(l1 ) (yl1 , . . . , yln , xd , . . . , x1 ) 0 M ξ, dyln+1 × λd (xd , . . . , x1 ) n tk ∧t R   + n Y × glj grj ylj dylj  ξ j=1

=M

Z ξ,

tk+1 ∧t

tk ∧t

∗ dy (Fl (tk )) Fr (tk ) ξ

where the last equality is achieved by Lemma 3.7 again. Let us suppose the result (l,r) (l,r) holds for any N tk ≤ N and prove it for N tk = N + 1. With ε0 defined as 0 0 n+1 in (3.11) and lj , rj j=1 the left-right index set uniquely determined by ε0 , since

334

VITONOFRIO CRISMALE

0 l10 < · · · < ln+1 , we have l10 is the index relative to the first annihilator moving from the right hand side. By the non-crossing arguments, it is easy to see that {l10 , r10 } o dε 0 ( l 0 ) n0 1 be is the first index consecutive pair from the right hand side. Let l h , r 0h h=1 0 0 n+1 the subset of lj , rj j=1 in which all the right indices are on the right hand side of l10 and r 0dε0 l 0 = r10 . If y is the variable relative to the operator A (tk ∧ t, tk+1 ∧ t) , ( 1)

then

ξ, A (tk ∧ t, tk+1 ∧ t) (Fl (tk ))∗ Fr (tk ) A+ (tk ∧ t, tk+1 ∧ t) ξ

= ξ, A (tk ∧ t, tk+1 ∧ t) · · · A gl10 A+ gr10 · · · A+ (tk ∧ t, tk+1 ∧ t) ξ

= |hξ, A (tk ∧ t, tk+1 ∧ t) A (g2n ) · · · ×   , yl10 , y, xd , . . . , x1 ) Z t λd+dε0 (l 0 ) +1 (yl 01 , . . . , yl 0d −1 1 0 ε0 (l )   1 g l10 gr10 yl10 dyl10  × × 0 0 0 , . . . , y , y c , y, x , . . . , x ) λ (y d 1 d+dε0 (l 0 ) l l1 0 l 1

1

d 0 0 −1 ε (l ) 1

E × · · · Aε(1) (g1 ) A+ (tk ∧ t, tk+1 ∧ t) ξ

Moreover, by Lemma 3.1, we obtain the quantity above is less than or equal to D M 2 ξ, A (tk ∧ t, tk+1 ∧ t) Aε(2n) (g2n ) · · · ×   , yl10 , yb, xd , . . . , x1 ) Z t λd+dε0 (l 0 ) (yl 01 , . . . , yl 0d −1 1 0 ε0 (l )   1 × g l10 gr10 yl10 dyl10  × 0 0 0 λ (y , . . . , y , y c , y b , x , . . . , x ) d+dε0 (l 0 ) −1 l d 1 l1 0 l 1

1

d 0 0 −1 ε (l ) 1

× · · · Aε(1) (g1 ) A+ (tk ∧ t, tk+1 ∧ t) ξ

E

Now in the sequence of operators on the right hand side of the scalar product above, we have exactly N index consecutive pairs. The induction hypothesis gives us the quantity is less than or equal to Z tk+1 ∧t

∗ 2 2N −1

M M ξ, ds Fl (tk ) Fr (tk ) ξ tk ∧t

and the thesis follows.

Before proving the last semi-martingale inequality, we introduce the following useful notation: (l,r) N tk := max N tk , N := max N tk 1≤l,r≤m

k=1,...,n

Proposition 3.11. Using the same notations as above, one has

2

Z t Z t

2 + 2N−1

kF (s) ξk ds F (s) dA (s) ξ ≤ M

0

(3.12)

0

Proof. In fact, using the adaptness arguments, the left hand side of (3.12) is equal to n X m X

ξ, αl αr A (tk ) (Fl (tk ))∗ Fr (tk ) A+ (tk ) ξ k=1 l,r=1


Using Lemma 3.10, one finds this is less than or equal to Z tk+1 ∧t Z t n X 2 2 kF (tk )k ξ ≤ M 2N−1 kF (s) ξk ds M 2N tk −1 ξ, tk ∧t

k=1

335

0

where we used the Cauchy-Schwarz inequality.

Let us consider the 1-mode type IFS case, i.e. the case (2.7). Fixed t ∈ [0, +∞) , any G ∈ Bt is represented by a sequence of creation and annihilation operators we need to know the number of. For example, if G = n Aε(n) (gn ) · · · Aε(1) (g1 ), where n ∈ N, g1 , . . . , gn ∈ L2 (0, t) , ε ∈ {−1, 1} , this number is equal to n. Since the representation of G is not unique, such a number runs over a set, whose minimum we call the order of G and denote by ordG. We recall that any F ∈ S can be written as F =

n X

F (tk ) χ[tk ,tk+1 ) , F (tk ) =

k=1

m X

αh Fh (tk ) , Fh (tk ) ∈ Btk

h=1

By definition ordFh (tk ) < +∞ for any tk , then we define (F )

Nt k

(F )

:= max ordFh (tk ) , N (F ) := max Ntk 1≤h≤m

1≤k≤n

For any d ∈ N let us take ωN (F ) +d+1 := max ωordFh (tk )+d+1 tk

1≤h≤m

ωN (F ) +d+1 := max ωN (F ) +d+1 1≤k≤n

where, as usual, for any n ∈ N, ωn := F+ (tk ) :=

λn λn−1

tk

and ω0 := 1. Moreover we put

H X (h,k) (h,k) c tk , f (h,k) A+ fh · · · A+ f1

h=0

F− (tk ) := f (h,k) :=

H X (h,k) (h,k) · · · A f1 c tk , f (h,k) A fh

h=0 (h,k) (h,k) fh , . . . f1

∈ L2 (0, t) , Ch . If for any d ≥ 1 Md := max {M1 , . . . , Md }

we find the following semimartingale estimates for left and right stochastic integrals of simple adapted processes. Proposition 3.12. Under the same notations of Proposition 3.2, for 1-mode type IFS one has:

Z t

2 Z t

2

≤ M2 F (s) dA (s) ξ kF (s) ηd k ds (3.13) d

0

0

2

Z t d−1 Z t X

≤ M2

dA (s) F (s) ξ kF− (s) ηd−h k2 ds

d 0

h=0

0

(3.14)

336

VITONOFRIO CRISMALE

Z t

2 Z t

2 +

dA (s) F (s) ξ ≤ MN (F ) +d+1 kF (s) ξk ds

0

(3.15)

0

Z t

2 Z t

2 +

≤ MN (F ) +d+1 kF+ (s) ξk ds F (s) dA (s) ξ

(3.16)

0

0

Proof. In fact, from (2.7)

2

2

Z t n

X

=

F (t ) A (t ∧ t, t ∧ t) ξ F (s) dA (s) ξ

k k k+1

0 k=1

2

Z tk+1 ∧t n

X

dxd gd (xd ) F (tk ) ηd = ωd

t ∧t k k=1

Z t

2

≤ Md2 dsgd (s) F (s) ηd

0

and (3.13) follows from the Cauchy-Schwarz inequality. Let us prove (3.14).

2

Z t

2 n

X

A (tk ∧ t, tk+1 ∧ t) F− (tk ) ξ dA (s) F (s) ξ

=

0

(3.17)

k=1

where the equality above follows from (2.11) and the adaptness of F (tk ) . If n X (h,k) (h,k) χ[tk ,tk+1 ) (r) · · · A f1 c tk , f (h,k) A fh

(h)

F− (r) :=

k=1

for any k = 1, . . . , n A (tk ∧ t, tk+1 ∧ t) F− (tk ) gd ⊗ · · · ⊗ g1   H∧(d−1) h E D Y X (h,k) , gd−j+1  × = ωd−j+1 fj c tk , f (h,k)  j=1

h=0

×

Z

tk+1 ∧t

ωd−h gd−h (r) dr gd−h−1 ⊗ . . . ⊗ g1

tk ∧t

Then n X

A (tk ∧ t, tk+1 ∧ t) F− (tk ) ξ

k=1

H∧(d−1)

=

X

h=0

ωd−h

Z

t 0

(h)

gd−h (r) F− (r) ηd−h dr


337

( h0 ) (h) As a consequence of orthogonality of F− (r) ηd−h and F− (r) ηd−h0 when h 6= h0 , the right hand side of (3.17) is equal to:

2 Z t H∧(d−1) X

(h)

ωd−h gd−h (r) F− (r) ηd−h dr

0

h=0

≤ Md2

d−1 Z X

h=0

t

2

kF− (r) ηd−h k dr

0

where in the last estimate we used the Cauchy-Schwarz inequality. For (3.15), by means of the usual adaptness arguments

Z t

2

+

dA (s) F (s) ξ

0

≤

= = ≤

n X

F (tk ) ξ, A (tk ∧ t, tk+1 ∧ t) A+ (tk ∧ t, tk+1 ∧ t) F (tk ) ξ k=1 n X

m X

k=1 l,r=1 n X m X k=1 l,r=1 n X

|αl αr | · Fl (tk ) ξ, A (tk ∧ t, tk+1 ∧ t) A+ (tk ∧ t, tk+1 ∧ t) Fr (tk ) ξ

Z ωordFr (tk )+d+1 |αl αr | · Fl (tk ) ξ,

ωN (F ) +d+1

k=1

tk

Z

tk+1 ∧t

2

tk+1 ∧t

tk ∧t

dr kF (tk ) ξk ≤ ωN (F ) +d+1

tk ∧t

drFr (tk ) ξ

Z

t

2

kF (s) ξk ds

0

Hence (3.15) follows. Finally, for (3.16), as a consequence of the adaptness of F (tk ), we have

2

Z t

2 n

X

+ +

F (s) dA (s) ξ = F (t ) A (t ∧ t, t ∧ t) ξ

+ k k k+1

0 k=1

which is equal to n X

ξ, A (tk ∧ t, tk+1 ∧ t) F− (tk ) F+ (tk ) A+ (tk ∧ t, tk+1 ∧ t) ξ k=1

Again by adaptness we have that for any k = 1, . . . , n the non zero contributions to the scalar product above can be obtained when A (tk ∧ t, tk+1 ∧ t) acts on A+ (tk ∧ t, tk+1 ∧ t) . It can be checked the quantity above is less than or equal to Z tk+1 ∧t n λ (F ) X Nt +d+1 λd k ξ, drF− (tk ) F+ (tk ) ξ · λd λN (F ) +d tk ∧t k=1 tk Z n tk+1 ∧t X drF− (tk ) F+ (tk ) ξ ≤ MN (F ) +d+1 ξ, tk k=1

≤ MN (F ) +d+1

Z

tk ∧t

t

2

kF+ (s) ξk ds

0

338

VITONOFRIO CRISMALE

where we used the Cauchy-Scwharz inequality. 4.

Stochastic Integral

In this section we extend the definition of stochastic integral to the vector space of processes that can be approximated by sequences of elements of S . We will follow the methods of [3] and [14] in order to set a definition of a stochastic integral satisfying our semimartingale inequalities. Let us take ξ = ud ⊗ . . . ⊗ u1 ∈ D and the set J (ξ) ⊂ D whose elements are Φ and uσ(h) ⊗ . . . ⊗ uσ(1) , h ∈ {1, ..., d} , σ : {1, ..., h} → {1, ..., d} increasing. As in [3] we want to establish a τ − semimartingale inequality with respect to a topology τ induced by a family of semi–norms. We recall that in [3] the topology τ is induced by the seminorms Z t 2 2 kF kξ,t,µ := kF (s) ξk dµ (s) , 0

where F is a simple adapted process, ξ ∈ D, t ∈ R+ are arbitrarily chosen. In our case, for any F ∈ S , ξ ∈ D , t ∈ R+ and for any N ≥ 1, the topology τ is determined by the seminorms n o qξ,t,N (F ) := max M 2 t, M 2N−1 , t × (Z 12 ) 12 Z t t X 2 2 ∗ kF (s) ηk ds + kF (s) ηk ds (4.1) × η∈J(ξ)

or

×

X

0

0

qξ,t,N (F ) (F ) := max Md2 , MN (F ) +d+1 , t × (Z 21 Z t 21 ) t X 2 2 ∗ + kFε (s) ηk ds kFε (s) ηk ds

η∈J(ξ) ε∈{−1,0,1}

0

(4.2)

0

according to whether we consider the general case, with non constant interacting functions, or the 1-mode type IFS, and F−1 = F− , F0 = F, F1 = F+ . From now on we will consider only the general case, as the 1-mode type IFS can be obtained just replacing (4.1) by (4.2). Denote by αβ an arbitrary element of the set {01, 10}. As a consequence of Proposition 3.2 and Proposition 3.11, the maps Z t F ∈ S 7→ F (s) dM αβ (s) ∈ L (D, FI ) 0

F ∈ S 7→

Z

t

dM αβ (s) F (s) ∈ L (D, FI ) 0

are continuous with respect to the topology on S induced by semi-norms (4.1) and the topology of strong ∗-convergence on D. Hence, denoting these topologies by τ and τ 0 respectively, we say that the basic processes are (τ − τ 0 ) -semimartingales, according to [3], Definition 2.1. Let S be the vector space of processes F such that there exists a sequence F (n) n≥0 in S for which the following property holds:


(∗) for any t ∈ R+ ∗-convergence on D.

F (n)

n≥0

339

converges to F (t) in the topology τ 0 of strong

The following definition gives us the class of integrable processes. Definition 4.1. A process F ∈ S is said to be integrable if there exists a sequence F (n) n≥0 in S satisfying the condition (∗) and such that for any ξ ∈ D, t ∈ R+ lim qξ,t,N F (n) − F = 0. (4.3) n→+∞

We denote by I the class of all integrable processes and notice that for any t ∈ R+ , any F ∈ I, and any (F n )n≥0 satisfying Definition 4.1, the sequences of stochastic integrals: Z t (n) αβ F (s) dM (s) Z

0

t

dM αβ (s) F (n) (s) 0

n≥0

n≥0

are convergent in the topology of strong ∗-convergence on D as a consequence of Propositions 3.2 and 3.11. Hence we can define the left and right stochastic integrals of elements of I with respect to the basic processes in the following way Z t Z t F (s) dM αβ (s) ξ := lim F (n) (s) dM αβ (s) ξ Z

0

n→+∞

t

dM αβ (s) F (s) ξ := lim 0

n→+∞

for any ξ ∈ D.

Z

0

t

dM αβ (s) F (n) (s) ξ, 0

Remark 4.2. In the construction of quantum stochastic integrals in both the cases (general and 1-mode type), we used the locally convex topology induced by the family of seminorms (4.1) or (4.2) and the topology of strong ∗-convergence on L (D, FI ). Nevertheless one can notice that it is possible to define a class of integrable processes and their quantum stochastic integrals by using only the strong ∗-convergence and the limit in the induced topology. The emergence of the other topology, together with condition (4.3), reveals in order that such integrals have some nice properties, allowing, for example, to give existence and uniqueness of the solution for a wide class of quantum stochastic differential equations, as we will se in [13]. The following result contains the semimartingale inequalities for the stochastic integrals of elements of I and will be useful in [13]. Proposition 4.3. For any F ∈ I, any

Z t

αβ

F (s) dM (s) ξ

≤ qξ,t,N (F ) , 0

ξ ∈ D, the following estimates hold:

Z t

αβ

dM (s) F (s) ξ

≤ qξ,t,N (F ) (4.4) 0

Proof. The thesis follows from Definition 4.1, Proposition 3.2 and Proposition 3.11.

340

VITONOFRIO CRISMALE

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23. Skeide, M.: Quantum stochastic calculus on full Fock modules; J. Funct. Anal. 173 (2000) 401-452. 24. Speicher, R.: Stochastic integration on the full Fock space with the help of a kernel calculus; Publ. Res. Inst. Math. Sci. 27 no. 1 (1991) 149-184. ` degli studi di Bari, Via E. Orabona, 4 Dipartimento di Matematica, Universita I-70125 Bari, Italy E-mail address: [email protected]