Semigroup Forum Vol. 68 (2004) 87–107
c 2003 Springer-Verlag New York Inc.
DOI: 10.1007/s00233-002-0024-2
RESEARCH ARTICLE
Perturbations of Bi-continuous Semigroups with Applications to Transition Semigroups on Cb (H) B´ alint Farkas Communicated by Rainer Nagel
Abstract We prove an unbounded perturbation theorem for bi-continuous semigroups on the space of bounded, continuous functions on the Hilbert space H . This is applied to the Ornstein-Uhlenbeck semigroup, thus providing a purely functional analytic approach to the existence of transition semigroups on Cb (H) with bounded non-linear drift. Key words and phrases: Unbounded perturbation, not strongly continuous semigroups, bi-continuous semigroups, equicontinuous semigroups, OrnsteinUhlenbeck semigroup, transition semigroup. 2000 Mathematics Subject Classification: 46A03, 47D06, 47D07, 47D99, 47A55.
1. Introduction The starting point of this investigation is the stochastic differential equation (SDE)
dX(t) = (AX(t) + F (X(t))) dt + Q1/2 dW (t), t ≥ 0 X(0) = x, x ∈ H.
In the spirit of Da Prato and his school, we investigate this equation by functional analytic methods (see [2], [5, 6, 7, 8], [9], [24], [25], [27] and [30]). In particular, we construct the transition semigroup on Cb (H) , H Hilbert space, associated to the solutions of (SDE) (the ingredients will be specified exactly in Section 4). Our approach is based on perturbation methods. We start from the Ornstein-Uhlenbeck operator generating a bi-continuous semigroup on Cb (H) , The author was partially supported by the Marie Curie Host Fellowship “Spectral theory for evolutions equations” at Arbeitsgemeinschaft Funktionalanalysis T¨ ubingen, contract number HPMT-CT-2001-00315 and by a DAAD-NATO fellowship.
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(cf. [20]). We then consider the perturbing operator Bf : =F, Df , which corresponds to the nonlinear drift term in (SDE). Such perturbations were studied by many authors. Investigations were θ already carried out on Cub (H) 1 and Cub (H) by G. Da Prato [5, 6] with dissipativity methods. He obtained C0 -semigroups on the closure of the domain of the Ornstein-Uhlenbeck operator L. The semigroup was also studied on Lp (H, µ) , µ an invariant measure (e.g., Da Prato [5]), and in particular on L2 (H, µ) by A. Rhandi [27]. A recent article identifies the domain of the Ornstein-Uhlenbeck generator L on Lp (RN ) and shows regularity properties of the corresponding semigroup [25]. Approximation methods were also applied as in [22] and [30], or in the framework of bi-continuous semigroups [1] when H is finite dimensional. In this paper, we show the existence of transition semigroups on the whole space Cb (H) for arbitrary drift term corresponding to F ∈ Cb (H, H) . Such results were obtained by Goldys and Kocan in the framework of strongly continuous semigroups with respect to mixed topologies [18]. In particular, Theorem 4.6 is a reformulation of Theorem 5.2 in [18]. The starting point is the paper by F. K¨ uhnemund [21], where she showed that the Ornstein-Uhlenbeck semigroup is a bi-continuous semigroup on Cb (H) . However, the perturbation theory of bi-continuous semigroups is not yet developed compared to that of C0 -semigroups. In the C0 -semigroup setting, e.g., when the Ornstein-Uhlenbeck semigroup is considered on L2 (H, µ) , perturbation theory turned out to be useful. A. Rhandi [27] used the Miyadera-Voigt perturbation theorem to show the existence of transition semigroups on L2 (H, µ) . So our main task is to prove an abstract perturbation theorem for bi-continuous semigroups applicable to operators B as above. In a previous paper, we proved a bounded perturbation theorem for general bi-continuous semigroups (see [17]). However, no unbounded perturbation results are known yet. In this work, we concentrate on the space Cb (Ω) endowed with the compact-open topology τc , and prove an unbounded perturbation theorem for bi-continuous semigroups on such spaces. This result is then applied to prove the existence of transition semigroups on the space Cb (H) for any bounded non-linear drift term F ∈ Cb (H, H) . As for the notation, we remark that Ω denotes a Polish space, while H always stands for a separable Hilbert space with scalar product ·, · and norm · . The Banach space Cb (Ω) is also equipped with another locally convex topology τc which corresponds to the locally-uniform convergence and 1
Cb stands for the space of bounded, continuous functions, Cub is the space of bounded, θ uniformly continuous functions, and Cub is the space of bounded, θ -H¨ older-continuous functions.
89
Farkas is determined by the family of seminorms P:=
pK : pK (f ) = sup |f (x)|, f ∈ Cb (Ω), K ⊆ Ω compact .
(1.1)
x∈K
The paper is organised as follows. In this introductory section we collect the basic properties of bi-continuous semigroups from the abstract theory as developed in [19] and [21] and, for the sake of completeness, we also repeat some parts of [17]. In Section 2, we study the space Cb (Ω) in detail (cf. also [18]) and obtain results allowing us to continue with Section 3. There we prove a Miyadera-Voigt type perturbation theorem for bi-continuous semigroups on Cb (Ω) . In the last section our perturbation result is then applied to show the existence of transition semigroups on Cb (H) . As a consequence, the theory immediately yields the tightness of the family of measures corresponding to these semigroups. Bi-continuous semigroups were introduced by F. K¨ uhnemund in [21], and we refer to [17], [19, 20] and [21] for the general theory, including generation, approximation or perturbation results. Here, we recall the notions that are needed in this paper. In this section X denotes a Banach space endowed with a locally convex topology τ determined by the family P of seminorms. We suppose that Assumption 1.1 of [21] is satisfied. We point out that Cb (Ω) fits into this framework (see [21, Sec.3.2]). Remark 1.1. Under the above assumptions the norm · is τ -lower semicontinuous, hence the unit ball in X is τ -closed. Proposition 1.2. Let T = (T (t))t≥0 be a bi-continuous semigroup with generator (A, D(A)). Then there exists η > 1 such that for all x ∈ X there exist xn ∈ D(A) with xn ≤ ηx Proof.
and
τ
xn → x.
(1.2)
For each x ∈ X , it is shown in [21, Sec. 1.2] that τ − lim nR(n, A)x = x. n→∞
Take M > 0 and ω ∈ R satisfying T (t) ≤ M eωt ,
t ≥ 0.
(1.3)
Since (A, D(A)) is a Hille-Yoshida operator, we have nR(n, A)x ≤
Mn x (n − ω)
for all n ∈ N .
Hence taking xn = nR(n, A)x and any η > M , we see that (1.2) is satisfied.
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If the previous proposition still holds for a set D replacing D(A) with an appropriate η > 1 , then we call it an η -bi-dense set. Therefore with this terminology, we have that D(A) is η -bi-dense for some η > 1 . As for measurability and integration, we may notice that for a τc continuous function F : R+ → X the map t → F (t) is lower semicontinuous, hence measurable. Further, if F is locally norm-bounded, then F ∈ L1loc (R+ ) . We will use these facts tacitly. We define the topology τA on D(A) by the set of seminorms PA : = {p(·) + p(A·) : p ∈ P} . It is clear that τA is finer than the restriction of τ to D(A) . Moreover, we endow D(A) with the graph norm xA : =x + Ax,
x ∈ D(A),
and denote this normed space by (D(A), · A ) or simply by D(A) . 2. Semigroups on Cb (Ω) In this section we restrict ourself to the space X = Cb (Ω) , where Ω is a Polish space, and we consider the compact-open topology τc on Cb (Ω) as in (1.1). Before proving our main perturbation theorem for such semigroups, we have to investigate sets of linear forms on Cb (Ω) . The dual of Cb (Ω) (as a Banach space) is isomorphic to the space M(βΩ) ˇ of all bounded, complex, Borel measures on the Stone-Cech compactification βΩ of Ω . Let J denote the set of norm-continuous linear functionals on Cb (Ω) which are also τc -continuous on norm-bounded sets. We show that J is indeed a norm closed subspace in M(βΩ) . Indeed, take ϕn ∈ J with ϕn → ϕ ∈ (Cb (Ω), · ) , and let xα ∈ Cb (Ω) be a norm-bounded net τc converging to x ∈ Cb (Ω) . We have to show that ϕ(xα ) → ϕ(x) . This however follows from |ϕ(x − xα )| ≤ |ϕ(x − xα ) − ϕn (x − xα )| + |ϕn (x − xα )| ≤ Kϕ − ϕn + |ϕn (x − xα )| ≤
ε 2
+
ε 2
= ε,
first by taking n ∈ N sufficiently large and then for fixed n using the continuity assumptions on ϕn . Obviously J is linear subspace. It is an interesting problem to find a characterisation of those measures in M(βΩ) belonging to J . To this end, we recall from [34, Ch. 1] that for a topologically complete space Ω , one always has that Ω is a Gδ set in βΩ . In particular, this is the case for a Polish space. Thus we can identify M(Ω) with a subspace of M(βΩ) by ι: M(Ω) → M(βΩ), [ι(ν)](B): =ν(Ω ∩ B)
for all ν ∈ M(Ω) and B ⊆ βΩ Borel set.
Farkas
91
Then ι is an injection with rg ι = {µ : µ ∈ M(βΩ), µ(βΩ\Ω) = 0} and ι(M(Ω)) ⊆ J . For the reverse inclusion take µ ∈ J and suppose that µ is positive. Assume that µ ∈ ι(M(Ω) ), i.e., µ(βΩ\Ω) > 0 . Since Ω is a Gδ , hence βΩ\Ω is an Fσ set, there exists a closed set F ⊆ βΩ with F ∩ Ω = ∅ and µ(F ) > 0 . Consider further a compact set K ⊆ Ω . This set is compact, thus closed in βΩ . Therefore, when F is a closed subset as considered previously, we have a continuous function fK,F on βΩ with values in [0, 1] satisfying fK,F (x) = 0 for all x ∈ K and fK,F (x) = 1 when x ∈ F . So we obtain fF,K , µ ≥ µ(F ) > 0. But this leads to contradiction by the arbitrariness of the compact set K and the τc -continuity of µ. After these preparations, consider a norm-bounded operator T on Cb (Ω) which is also τc -continuous on norm-bounded sets. By the above characterisation of M(Ω) , its adjoint T ∈ L(M(βΩ)) leaves M(Ω) invariant. We recall that a set K ⊆ M(Ω) is said to be weak ∗ -compact 2 if it is compact for the topology σ(M(Ω), Cb (Ω)) . Lemma 2.1. Let T : R+ → L(Cb (Ω)) be a τc -strongly continuous function and K ⊆ M(Ω) a weak ∗ -compact set. Then the map [0, +∞) × K (t, ν) → T (t)ν ∈ M(βΩ) is continuous if we take the weak ∗ topology σ(M(βΩ), Cb (Ω)) on K and on M(βΩ) . Proof.
Let t ∈ [0, +∞) and ν ∈ K fixed. Then
|f, T (t)ν − T (t )ν | = |T (t)f, ν − T (t )f, ν | ε ε |T (t)f, ν − ν | + |T (t)f − T (t )f, ν | ≤ + for all f ∈ Cb (Ω) . 2 2 The last inequality follows by the continuity assumptions and by Prokhorov’s theorem 3 for t ∈ U and ν ∈ V with appropriate neighbourhoods U, V of t and ν . Lemma 2.2. Let T : [0, +∞) → L(Cb (Ω)) be a τc -strongly continuous semigroup. For a weak ∗ -compact set K ⊆ M(Ω) and t0 > 0 the set of measures {T (t)ν : t ∈ [0, t0 ], ν ∈ K} is σ(M(βΩ), Cb (Ω))-compact. 2 In
probability theory, sometimes the term weakly-compact is used. e.g., A. N. Shiryayev [29, Sec. III.2].
3 See
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Proof. It is a straightforward consequence of Lemma 2.1 and the fact that the set [0, t0 ] × K is compact for the product topology. We now reformulate the previous results using the concept of tightness from measure theory (cf. [29, Sec. III.2]). Lemma 2.3. Let T : [0, +∞) → L(Cb (Ω)) be a τc -strongly continuous semigroup consisting of operators that are τc -continuous on norm-bounded sets. For a norm-bounded, weak ∗ -compact set K ⊆ M(Ω) and t0 > 0 the set of measures {T (t)ν : t ∈ [0, t0 ], ν ∈ K} is tight. Proof. The assertion follows immediately from Lemma 2.2, taking into account the arguments preceding Lemma 2.1 concerning the invariance of M(Ω) . As a final result we show that bi-continuous semigroups on Cb (Ω) of linear operators that are τc -continuous on norm bounded sets satisfy a property which might be called “local”. This proposition will be essential in the proof of the perturbation result (cf. Theorem 3.2). Theorem 2.4. Let T be a bi-continuous semigroup on Cb (Ω) , and assume that T (t) is τc -continuous on norm-bounded sets for all t ≥ 0 . Then for all t0 ≥ 0 , K ⊆ Ω compact and ε > 0 there exists a constant MK,ε > 0 and a compact set K ⊆ Ω such that sup |T (t)f (x)| ≤ MK,ε sup |f (x)| + εf
(EQ)
x∈K
x∈K
for all t ∈ [0, t0 ] and f ∈ Cb (Ω). Proof. Let ε > 0 , t0 > 0 and K ⊆ Ω be a compact set. Take a compact set K ⊆ Ω such that [T δx ](Ω\K ) ≤ ε for all t ∈ [0, t0 ]. Such compact set exists by Lemma 2.3 (cf. also [3]). We then obtain sup |T (t)f (x)| = sup f dT (t)δx x∈K
x∈K
≤ sup x∈K
≤
Ω
K
|f | d|T (t)δx | + sup x∈K
|f | d|T (t)δx |
Ω\K
sup T (t) · sup |f (x)| + εf , t∈[0,t0 ]
x∈K
(2.1)
which is the assertion. Remark 2.5. It is easy to see that (EQ) means exactly the equicontinuity of {T (t) : t ∈ [0, t0 ]} with respect to the mixed topology τ0M investigated in
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[18]. This, in particular, gives that the class of semigroups that are strongly continuous and locally equicontinuous with respect to the mixed topology τ0M and the class of bi-continuous semigroups that satisfy (EQ) coincide. Moreover, by [18, Proposition 2.3], it follows that bi-continuous semigroups on (Cb (Ω), τc ) are the same as τ0M strongly continuous and locally sequentially-equicontinuous semigroups. On the other hand, from [28, Section 8 and 9], it follows that whenever Ω is a Polish space the sequential τ0M -continuity is the same as the τ0M -continuity of a linear operator T0 on Cb (Ω) 4 . Combining these results, we have that every bi-continuous semigroup on Cb (Ω) , Ω Polish space, satisfies the inequality (EQ) (cf. also [13, 14, 15]). It is however not known to the author whether an analogous estimate to (EQ) is valid for general bi-continuous semigroups. 3. Miyadera-Voigt perturbations In this section, we prove a perturbation result for bi-continuous semigroups on (Cb (Ω), τc ). The result and the method is inspired by the analogous result for C0 -semigroups (see [16, Sec. III.3.c] and [33]). We use the same techniques as in [16] or [17] and define first an abstract Volterra operator by
t
F (t − s)BT (s)f ds for each f ∈ D(A) .
[VF ](t)f : = 0
However, we would like to prove an unbounded perturbation theorem, so B (and hence [V F ](t) ) is generally not everywhere defined. Even worse is that in general B is not defined on a dense linear subspace, thus to obtain a continuous extension of [V F ](t) to the whole space Cb (Ω) requires more subtle arguments. The proof is more involved than in [17] and Theorem 2.4 plays a key role in the reasonings. We introduce the following space for t0 > 0 Xt0 : ={F : [0, t0 ] → L(Cb (Ω)) τc -strongly continuous, norm-bounded, and {F (t) : t ∈ [0, t0 ]} satisfies (EQ)}. It is clear that Xt0 is a linear space, and a similar proof as in [17, Lemma 3.3] shows that Xt0 is a Banach space. Lemma 3.1. Let (T (t))t≥0 be a bi-continuous semigroup on Cb (Ω) with generator (A, D(A)), F ∈ Xt0 , B: (D(A), τA ) → (Cb (Ω), τc ) be continuous on · A -bounded sets and suppose that for all f ∈ D(A) the orbits t → BT (t)f, 4 Note
t ∈ [0, t0 ]
that in [28] the mixed topology is denoted by β0 and is called the substrict topology.
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are norm-bounded. Then the functions t → F (t)BT (t)f,
t ∈ [0, t0 ],
are continuous for the topology τc . Proof. First of all, we remark that t → BT (t)f is τc -continuous for all f ∈ D(A) . Take now 0 < t < t0 and hn ∈ R converging to 0 . For f ∈ D(A) we write F (t + hn )BT (t + hn )f − F (t)BT (t)f = F (t + hn )BT (t + hn )f − F (t + hn )BT (t)f + F (t + hn )BT (t)f − F (t)BT (t)f. Clearly we have τ
c F (t + hn )BT (t)f − F (t)BT (t)f → 0.
From the bi-equicontinuity of {F (t) : t ∈ [0, t0 ]} and the boundedness of {BT (t) : t ∈ [0, t0 ]} it follows also that F (t + hn )BT (t + hn )f − F (t + hn )BT (t)f τ
c = F (t + hn )(BT (t + hn ) − BT (t))f → 0.
The left (respectively the right) τc -strong continuity of F (·)BT (·) in the endpoints of [0, t0 ] can be proved analogously. Theorem 3.2. Let T be a bi-continuous semigroup on Cb (Ω) . Denote its generator by (A, D(A)) and suppose that D(A) is η -bi-dense in Cb (Ω) . Further, let B: (D(A), τA ) → (Cb (Ω), τc ) be continuous on · A -bounded sets. Suppose that there exist t0 > 0 and a real number q < 1/η 2 such that i) s → BT (s)f is bounded on [0, t0 ] for all f ∈ D(A), ii) for all t ∈ [0, t0 ] and f ∈ D(A)
t
BT (s)f ds < qf , 0
iii) for all norm-bounded sequence fn ∈ D(A), ε > 0 and for all p ∈ P there exists a seminorm p ∈ P and a positive real number Mp,ε (both possibly dependent on the norm bound of fn ) such that 0
for all n ∈ N .
t0
p(BT (s)fn ) ds < Mp,ε p (fn ) + εfn
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Farkas Then (A + B, D(A)) generates a bi-continuous semigroup S .
Proof. Let B be as specified in the assertion. Then for arbitrary f ∈ D(A) and F ∈ Xt0 the orbit s → F (t − s)BT (s)f is τc -continuous according to Lemma 3.1. This implies that the Riemann integral below makes sense and defines a linear operator B(F, t) B(F, t)f : =
t
F (t − s)BT (s)f ds for each f ∈ D(A) .
(3.1)
0
¯ t) on Cb (Ω) which is We show that B(F, t) extends to linear operator B(F, τc -continuous on norm-bounded sets for all t ∈ [0, t0 ], F ∈ Xt0 . First, let f ∈ Cb (Ω) and fn ∈ D(A) be an arbitrary sequence which is τc -convergent to f and satisfies fn ≤ K for some K > 0 . Then we have for p ∈ P and F ∈ Xt0
t
p(B(F, t)(fn − fm )) = p ≤
F (t − s)BT (s)(fn − fm ) ds
0 t
p(F (t − s)BT (s)(fn − fm )) ds 0
≤ KF,p,ε
t
p (BT (s)(fn − fm )) ds
0 t
BT (s)(fn − fm ) ds.
+ε 0
By making use of assumption ii), we may write p(B(F, t)(fn − fm )) ≤ KF,p,ε · Mp ,ε · p (fn − fm ) + (ε · KF,p,ε + εq)fn − fm ≤ KF,p,ε · Mp ,ε · p (fn − fm ) + 2K(ε · KF,p,ε + εq). This shows that taking first ε > 0 very small and then choosing ε > 0 appropriately small, p(B(F, t)(fn − fm )) can be made sufficiently small for large indices n and m, i.e., we see that B(F, t)fn is a τc -Cauchy sequence. Since it is also norm-bounded, by the sequential completeness we obtain that this sequence ¯ t)f . One sees immediately that is convergent, and we denote its limit by B(F, ¯ t)f is indeed independent of the particular choice of the the definition of B(F, ¯ t) is linear. It is also important to note that sequence fn and further that B(F, according to this definition, we have ¯ t)f ) ≤ KF,p,ε · Mp ,ε · p (f ) + K(ε · KF,p,ε + εq). p(B(F,
(3.2)
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Moreover, if we choose fn as in (1.2), we see that ¯ t)f ) ≤ KF,p,ε · Mp ,ε · p (f ) + ηf · (ε · KF,p,ε + εq), p(B(F,
(3.3)
thus ¯ t)f ) ≤ CF,p,ε · p (f ) + ε · f , p(B(F,
(3.4)
¯ t) with an appropriate constant CF,p,ε . We now proceed by showing that B(F, is norm-bounded. Indeed, take η > 0 as in (1.2) and let f ∈ Cb (Ω) arbitrary. By the assumption there exists fn ∈ D(A) such that (1.2) holds. We can now estimate the norm as B(F, t)fn =
sup
φ∈(Cb (Ω),τc ) φ≤1
≤
t F (t − s)BT (s)f ds, φ n 0
φ∈(Cb (Ω),τc ) φ≤1
≤
sup
φ∈(Cb (Ω),τc ) φ≤1
t
|F (t − s)BT (s)fn , φ| ds
sup
0
F ∞ φ
t
BT (s)fn ds ≤ qF ∞ fn 0
≤ ηqF ∞ f .
(3.5)
¯ t)f , we Hence B(F, t)fn is norm-bounded. Since it is τc -convergent to B(F, have by Remark 1.1 that ¯ t)f ≤ ηqF ∞ · f , B(F, ¯ t) ∈ L(Cb (Ω)) . In particular, it follows that B(F, ¯ ·) is a showing that B(F, bounded function. ¯ ·) ∈ Xt . From (3.4) we see that B(F, ¯ ·) satisfies We now prove that B(F, 0 ¯ (EQ), in particular, {B(F, t) : t ∈ [0, t0 ]} is bi-equicontinuous. ¯ t) on [0, t0 ] . Indeed, It is easy to see the τc -strong continuity of t → B(F, let f ∈ D(A) arbitrary. Then p((B(F, t + h) − B(F, t))f )
t t+h = p F (t + h − s)BT (s)f ds − F (t − s)BT (s)f ds 0
≤ p
t
0
(F (t + h − s) − F (t − s))BT (s)f ds
0
t+h
F (t + h − s)BT (s)f ds
+p t
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t
≤
p((F (t + h − s) − F (t − s))BT (s)f ) ds 0
t+h
p(F (t + h − s)BT (s)f ) ds → 0
+ t
as h → 0 , since in the first term the integrand p((F (t+h−s)−F (t−s))BT (s)f ) is dominated by F ∞ · BT (s)f and converges to 0 for all s ∈ [0, t] . The second term obviously converges to 0 . The case when h < 0 can be handled similarly. For an arbitrary f ∈ Cb (Ω) let ε > 0 and p ∈ P and take a sequence τc fn → f which is also norm bounded and satisfies the estimate in (1.2). Then ¯ t + h)f − B(F, ¯ t)f ) p(B(F, ¯ t + h)(f − fn ) + B(F, ¯ t + h)fn − B(F, ¯ t)fn + B(F, ¯ t)(fn − f )) = p(B(F, ¯ t + h)(f − fn )) + p(B(F, ¯ t + h)fn − B(F, ¯ t)fn ) ≤ p(B(F, ¯ t)(fn − f )) ≤ ε + p(B(F, holds for sufficiently small h > 0 , which can be concluded by means of the ¯ t) : t ∈ [0, t0 ]} and choosing n sufficiently bi-equicontinuity of the family {B(F, large. ¯ t) ∈ Xt , and also that It is now straightforward to conclude that B(F, 0 the Volterra operator V = Vt0 given by ¯ t)f (Vt0 F )(t)f : = B(F, is a bounded operator on the Banach space Xt0 . Moreover, for the operator norm we have that [Vt0 F ](t)f ≤ ηqF ∞ · f , hence Vt0 < 1/η.
(3.6)
By this fact, it follows that 1 ∈ .(V) . Let t > 0 be arbitrary. We then write t = nt0 + t1 , where n ∈ N , t1 ∈ [0, t0 ) and define S(t) : = (δt0 (R(1, V)T [0,t0 ] ))n · δt1 (R(1, V)T [0,t0 ] ). This means particularly that (IE)
t
S(t − s)BT (s)f ds
S(t)f = T (t)f + 0
for all f ∈ D(A) and t ∈ [0, t0 ] . We show that S: [0, +∞) → L(Cb (Ω)) is a semigroup. First, take 0 ≤ s, t ≤ s + t ≤ t0 .
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Analogously to the norm-bounded case [17] (cf. also [16, Sec. III.3.c]), we can show that S(t + s) = S(t)S(s). (3.7) Second, take any t, s > 0 and write these real numbers in the form t = nt0 + t1 ,
s = mt0 + t2 ,
n, m ∈ N , t1 , t2 ∈ [0, t0 ).
We then have S(t)S(s) = S(t0 )n · S(t1 ) · S(t0 )m · S(t2 ) = S(t0 )n · S(t1 ) · S(t0 ) · S(t0 )m−1 · S(t2 ) = S(t0 )n · S(t1 ) · S(t0 − t1 ) · S(t1 ) · S(t0 )m−1 · S(t2 ) · · · · · · = S(t0 )n · S(t0 )m · S(t1 ) · S(t2 ), but S(t0 )n · S(t0 )m · S(t1 ) · S(t2 ) if t1 + t2 < t0 , S(t0 )n+m · S(t1 + t2 ) = S(t0 )n+m+1 · S(t2 − (t0 − t1 )) if t1 + t2 ≥ t0 , which in both cases equals S(t + s) by definition. It is also clear that S(0) = I , hence S is a semigroup. Since we have S [0,t0 ] = R(1, V)T [0,t0 ] , so S is locally-bounded and {S(t) : t ∈ [0, t0 ]} is bi-equicontinuous and satisfies (EQ). Let t > 0 be given, we show that {S(s) : s ∈ [0, t]} is bi-equicontinuous. Indeed, let m = t/t0 . Since {S(t0 )k : k = 1, . . . , m} is τc -equicontinuous on norm-bounded sets, it is immediate that {S(t0 )k : k = 1, . . . , m} · {S(s) : s ∈ [0, t0 ]} is also bi-equicontinuous. Furthermore, (EQ) is satisfied. The τc -strong continuity of S is also obvious from the semigroup property (3.7), since S [0,t0 ] is τc strong continuous by definition. Thus we have proved that S is a bi-continuous semigroup. Now, we prove that the generator of S is (A + B, D(A)) . As a first step, we show that the resolvent set of (A + B, D(A)) is not empty. Let λ ∈ .(A). Taking the Laplace transform of T , we obtain for all f ∈ D(A) that R(λ, A)f =
∞ k=0
e−λkt0
0
t0
e−λt T (t)T (kt0 )f dt,
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where the series and the integral converges in the topology τc and, in this case also in the topology τA (cf. [16, Sec. III.3.c]). Hence we also have BR(λ, A)f =
∞
−λkt0
∞
t0
e
e−λt BT (t)T (kt0 )f dt
0
k=0
=
e−λkt0 [Vt0 Fλ ](t0 )T (kt0 )f
k=0
for all f ∈ D(A) , where Fλ = e−λ(t0 −·) I . This yields the norm estimate BR(λ, A)f ≤
∞
e−λkt0 [Vt0 Fλ ](t0 )T (kt0 )f
k=0
≤
∞
e−λkt0 Vt0 · Fλ · T (kt0 ) · f
k=0
≤ Vt0 · f + M Vt0
∞
e−λkt0 +ωkt0 f
k=1
≤ Vt0 · f +
(ω−λ)t0
e M · f η 1 − e(ω−λ)t0
for f ∈ D(A) .
Now, let f ∈ Cb (Ω) arbitrary and take any sequence fn ∈ D(A) as in Proposition 1.2. In this case BR(λ, A)fn converges to BR(λ, A)f in the topology τc . So we have by Remark 1.1 that BR(λ, A)f ≤ lim sup BR(λ, A)fn n→∞
M e(ω−λ)t0 ≤ lim sup Vt0 · fn + f · n η 1 − e(ω−λ)t0 n→∞ ≤ ηVt0 · f + M
e(ω−λ)t0 f . 1 − e(ω−λ)t0
From this we obtain that BR(λ, A) < 1 for sufficiently large λ . For such λ we have −1 R(λ, A + B) = R(λ, A) (I − BR(λ, A)) , showing .(A + B) = ∅ . Now, take f ∈ D(A) and t < t0 . Then by (IE) we can write S(t)f − f T (t)f − f 1 t S(t − s)BT (s)f ds. = + t t t 0 It is straightforward that this quotient remains bounded as t → 0 and that the first term converges to Af in the topology τc by definition. By the τc continuity of the orbits s → S(s)Bf one can choose 0 < δ < t0 for a given
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ε > 0 such that p(S(t − s)Bf − Bf ) ≤
ε 2
whenever t ∈ [0, δ] and s ∈ [0, t] . Further, by taking δ possibly smaller, we can write by the bi-equicontinuity of {S(s) : s ∈ [0, t0 ]} and the τc -continuity of s → BT (s)f that t 1 p S(t − s)BT (s)f ds − Bf t 0 1 t 1 t ≤ p(S(t − s)Bf − Bf ) ds + p(S(t − s)(BT (s)f − Bf )) ds t 0 t 0 ε ε ≤ + = ε. 2 2 This shows that A+B is a restriction of the generator C of S , that is A+B = C indeed. If T is a quasi-contractive, bi-continuous semigroup, i.e., if we can take M = 1 in (1.3), then η > 1 can be arbitrary, hence in (3.6) Vt0 < 1 replaces Vt0 < 1/η . As a consequence, it also suffices to require q < 1 in Theorem 3.2. 4. Application Let (A, D(A)) be the generator of the C0 -semigroup S on a separable Hilbert space H , and consider an H -valued cylindrical Wiener process W . Let Q be a bounded, positive, self-adjoint operator on H with trivial kernel. Define Q(t) =
t
S(s)QS ∗ (s) ds,
0
where the integral is understood in the strong sense. Assume additionally that Q(t) is of trace class for all t ≥ 0 . Now, consider the stochastic differential equation (see [11] for details)
dX(t) = (AX(t) + F (X(t))) dt + Q1/2 dW (t) t ≥ 0 X(0) = x x ∈ H.
Then the solutions X(t, x) give rise to linear operators R(t) forming a semigroup R on the space of bounded measurable functions [R(t)f ](x): = E[f (X(t, x))]
for all t ≥ 0 .
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Our goal is to obtain this semigroup as a perturbation of the Ornstein-Uhlenbeck semigroup P which corresponds to the equation dX(t) = AX(t) dt + Q1/2 dW (t), t ≥ 0, X(0) = x, x ∈ H. For all x ∈ H and t ≥ 0 there exist Gaussian measures N (S(t)x, Q(t)) with mean S(t)x and covariance operator Q(t) (cf. [8, 23] and [31]). The OrnsteinUhlenbeck semigroup P on Cb (H) is then given by the following formula ([8, 21]) P (t)f (x) = f dN (S(t)x, Q(t)) for all f ∈ Cb (H), x ∈ H , t > 0 . H
P (0) = I This semigroup and its properties have been investigated by many authors, see e.g., [2], [5, 6, 7, 8], [9], [10, 11], [18], [26] and [32]. Furthermore, it is proved in [21] that it is a bi-continuous semigroup for the compact open topology τc on Cb (H) (cf. also [18]). Hence, this semigroup fits into our setting, and we can argue in the spirit of [27], applying our Miyadera-Voigt type perturbation result, i.e., Theorem 3.2. We denote the generator of P (t) as a bi-continuous semigroup by (L, D(L)) . Remark 4.1. From the definition of P (t) , it is immediate that P (t) is τc continuous on norm bounded sets of Cb (H) . Thus Theorem 2.4 applies, and we see that P (t) satisfies (EQ). Taking the Laplace transform of P (t) , we obtain the estimate for λ > 0 , ε > 0 and K ⊆ H compact +∞ sup |R(λ, L)f (x)| ≤ sup e−λs |P (s)f (x)| ds x∈K
x∈K
0
t
≤ sup x∈K
t
= sup x∈K
≤ 0
t
e−λs |P (s)f (x)| ds +
+∞
e−λs |P (s)f (x)| ds
t
+∞
e−λs f ds
t
0
x∈K
0
≤ sup x∈K
e−λs |P (s)f (x)| ds + sup
t
e−λs |P (s)f (x)| ds +
0
e−λt f λ
e−λs ( sup |f (x)| + ε f ) ds + x∈K
≤
1 e−λt ( sup |f (x)| + ε f ) + f λ x∈K λ
≤
1 sup |f (x)| + εf λ x∈K
for a suitable K ⊆ H compact set.
e−λt f λ
(4.1)
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We now collect important properties of P from [8]. Denote by Df the derivative of a Fr´echet differentiable function f ∈ Cb (H) . In the sequel, we assume that rg S(t) ⊆ rg Q1/2 (t)
(∗)
and introduce the following function Γ(t): =Q(t)−1/2 S(t)
for t > 0 .
The next result is basically proved in [8] as Proposition 2.3. It is, however, valid for bounded, continuous functions as presented here. Lemma 4.2.
For all f ∈ Cb (H) and t > 0 we have P (t)f ∈ Cb1 (H) and
Γ(t)h, Q(t)−1/2 ·f (S(t)x + ·) dN (0, Q(t)),
[DP (t)f ](x), h =
(4.2)
H
and hence |[DP (t)f ](x)|2 ≤ Γ(t)2 · |[P (t)f 2 ](x)|
for all x ∈ H .
(4.3)
In particular, |[DP (t)f ](x)| ≤ Γ(t) · f
for all x ∈ H .
(4.4)
From now on, we shall assume the following (∗∗)
t0
Γ(s) ds < +∞ for some t0 > 0 .
0
The next result is similar to [8, Proposition 2.4]. Lemma 4.3.
For λ > 0 and f ∈ Cb (H) we have R(λ, L)f ∈ Cb1 (H), and [DR(λ, L)f ](x) ≤ M · f ,
(4.5)
where M is independent of f . Moreover, for all K ⊆ H compact set and ε > 0 there exist a constant MK,ε > 0 and a compact set K ⊆ H such that sup [DR(λ, L)f ](x) ≤ MK,ε · sup |f (x)| + εf . x∈K
Proof.
x∈K
Let t0 > 0 be as in (**) and 0 < λ ∈ .(L) . Then R(λ, L)f = e−λt0 P (t0 )R(λ, L)f +
0
t0
e−λs P (s)f ds
(4.6)
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Farkas for all f ∈ Cb (H) . By (4.3) we have [DR(λ, L)f ](x) ≤ e−λt0 [DP (t0 )R(λ, L)f ](x) +
t0
e−λs [DP (s)f ](x) ds
0 −λt0
≤ e
Γ(t0 ) · |[P (t0 )(R(λ, L)f ) ](x)|1/2 2
t0
+
Γ(s) · |[P (s)f 2 ](x)|1/2 ds.
0
From this, we conclude [DR(λ, L)f ](x) ≤
−λt0
e
Γ(t0 ) · R(λ, L) +
t0
Γ(s) ds f
0
showing (4.5). Further, by Remark 4.1 we can write sup [DR(λ, L)f ](x) ≤ e−λt0 Γ(t0 )( sup |(R(λ, L)f )(x)| x∈K
x∈K
+ εR(λ, L) · f ) t0 + Γ(s) ds · ( sup |f (x)| + εf ) x∈K
0
≤ e−λt0 Γ(t0 ) 1 × sup |f (x)| + εf + εR(λ, L) · f λ x∈K t0 + Γ(s) ds · ( sup |f (x)| + εf ) x∈K
0
≤ MK,ε sup |f (x)| + M εf . x∈K
This justifies (4.6) and finishes the proof. We are now ready to state our main result. Let F ∈ Cb (H, H) and define the operator B by 1
D(B) : = Cb (H), Proposition 4.4.
(Bf )(x) : = F (x), Df (x).
Let B be as above. Then for all f ∈ D(L) the orbits t → BP (t)f,
t ≥ 0,
are locally norm-bounded. Proof.
By Lemma 4.3 one sees that B ∈ L(D(L), Cb (H)). Since the set {P (t)f : t ∈ [0, t0 ]}
is · L bounded for all f ∈ D(L) , the assertion follows immediately.
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Proposition 4.5. Let B be as above. Then B: (D(L), τL ) → (Cb (H), τc ) is continuous on · L norm-bounded sets. In particular the orbits t → BP (t)f,
t ≥ 0,
are τc -continuous for all f ∈ D(L). Proof. The continuity of B is immediate from (4.6). Also the continuity of the orbits follows from this and Proposition 4.4. Notice that Proposition 4.4 shows assumption i) in Theorem 3.2 is fulfilled. From (4.4) and (4.5) we conclude that ii) also holds for a sufficiently small t0 > 0 . Assumption iii) in Theorem 3.2 is justified by the integrated version of (4.3). Putting together these results and applying Theorem 3.2 we obtain the following. Theorem 4.6. Let P be the Ornstein-Uhlenbeck semigroup on Cb (H) with generator L. Define the operator B as 1
D(B) : = Cb (H),
(Bf )(x): =F (x), Df (x).
Then (L+B, D(L)) is the generator of a bi-continuous semigroup R on Cb (H). These semigroups are referred to as transition semigroups, and Section 2 immediately yields the following corollary. Corollary 4.7. Let R be a transition semigroup on Cb (H) as obtained previously and K ⊆ M(H) be a norm-bounded, weak ∗ -compact set. Then the family {R (t)ν : t ∈ [0, t0 ], ν ∈ K} is tight. In particular, for all K ⊆ H compact set we have the tightness of the family {R (t)δx : t ∈ [0, t0 ], x ∈ K}. Remark 4.8. Our approach also allows different kinds of perturbing operators than B as above. Such results will be presented in a forthcoming paper where we establish also the positivity of the perturbed transition semigroups. Acknowledgment The author is beholden to A. Rhandi, E. Sikolya and A. Es–Sarhir for fruitful discussions and helpful suggestions. We thank the anonymous referee for his constructive comments. The financial support of the DAAD and the European Union is gratefully acknowledged.
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Mathematisches Institut Universit¨ at T¨ ubingen Auf der Morgenstelle 10 D-72076 T¨ ubingen, Germany
[email protected]
Received November 10, 2002 and in final form March 15, 2003 Online publication August 15, 2003
Department of Mathematics and its Applications Central European University Budapest
[email protected]
