Nonlinear Gaussian Transformation in Hilbert Spaces ⋆

Nonlinear Gaussian Transformation in Hilbert Spaces ? Agnieszka Kozak and Yuri Kozitsky Instytut Matematyki, Uniwersytet Marii Curie-SkÃlodowskiej PL 20-031 Lublin (Poland)

Key words: Gaussian measure; infinite dimensional holomorphy; holomorphic mapping; Fr´echet space; elliptic operator; Cauchy problem; fixed point; stability; hierarchical model

1

Introduction and Motivating Examples

Throughout H, H0 stand for real separable Hilbert spaces with the scalar products (·, ·), (·, ·)0 , norms | · |, | · |0 and σ-algebras of Borel subsets B, B0 . We write N0 for N ∪ {0}. Let γS be a symmetric Gaussian measure on H, which is uniquely determined by Z H

½

¾

1 exp {i(x, y)} γS (dy) = exp − (Sx, x) , 2

(1.1)

where a linear strictly positive trace class operator S : H → H is its covariance operator. For an appropriate function f : H → R, the mapping Z

f 7→ S(f ),

S(f )(x) =

f (x + y)γS (dy),

(1.2)

H

is a Gaussian integral transformation (S-transformation), see e.g. [4]. We introduce and study the following nonlinear generalization of (1.2) f 7→

Gσ,θ m,S (f ),

Gσ,θ m,S (f )(x)

=

Z h

√ im f (σx + σ θy) γS (dy),

(1.3)

H

? Supported in part by KBN Grant 2P03A 02915

Preprint submitted to Elsevier Science

4 July 2002

where m ∈ N \ {1}, θ ≥ 0 and σ > 0 are the parameters. As domain and range spaces of Gσ,θ m,S we use certain spaces of holomorphic functions. The motivation σ,θ to study Gm,S may be based on the following examples. Let f : H → R be a twice differentiable function on H. We define an elliptic operator [5] as follows 1 L(f )(x) = trace{Sf 00 (x)}, 2

(1.4)

where S : H → H is a strictly positive trace class operator. Consider the Cauchy problem ∂ft = L(ft ), ∂t

def

f0 = g : H → R,

t ∈ R+ = [0, +∞).

(1.5)

By Theorem 3.1 of [5], this problem has a unique solution for t ∈ [0, T ) ⊂ R+ , if g satisfies corresponding conditions. This solution may be written as Z

ft (x) =

g(x +

√

ty)γS (dy).

H

As in [9] one may modify the evolution described by (1.5) as follows. Let us divide the time half-line R+ into the intervals [(n − 1)θ, nθ], n ∈ N with some θ > 0. On each such an interval the evolution is to be governed by (1.5) but at the endpoints t = nθ, n ∈ N the function ft is changed impulsively fnθ+0 (x) = [fnθ−0 (σx)]m ,

σ > 0,

m ∈ N \ {1}.

Instead of considering ft defined on the whole R+ , one may study the sequence (n) (n) of functions {ft }n∈N0 , where ft = fθ(n+t) and t ∈ [0, 1]. Then this sequence ought to satisfy the following modified problem (see [7], [9] for more details) (n)

∂ft (n) = θL(ft ), t ∈ [0, 1]; ∂t h im (n) (n−1) f0 (x) = f1 (σx) , n ∈ N,

(1.6) (0)

f1 = g.

Its solutions may be obtained recursively by means of (1.3) as follows (n)

ft

(n−1)

= Gσ,tθ m,S (f1

),

(0)

f1 = g.

Another problem where transformations of the type of (1.3) may appear is the description of many component systems with hierarchical structures, such as models of hierarchically interacting quantum anharmonic oscillators [1] - [3], 2

interacting diffusions and models of population genetics [6]. A system of this kind, close to that studied in [1] - [3], may be described as follows. Let M(H0 ) stand for the set of symmetric probability measures µ on H0 such that Z

exp(a|x|20 )µ(dx) < ∞,

(1.7)

H0

with a certain a > 0. For such a measure, µ?m , m ∈ N will denote the convolution of its m copies. For nonnegative θ, δ and m ∈ N \ {1}, we set !

Ã

1 θ 2 ?m (1+δ)/2 dx), exp |x| µ (m Zµ 2 0 Ã ! Z θ 2 ?m (1+δ)/2 Zµ = exp |x| µ (m dx), 2 0

Nθ,δ m (µ)(dx) =

(1.8)

H0

where for a certain α > 0 and a suitable function f , we write Z

Z

f (α−1 x)µ(dx).

f (x)µ(αdx) =

By means of (1.8), one may generate sequences of measures {µn }n∈N0 , where µn = Nθ,δ m (µn−1 ),

µ0 ∈ M(H0 ),

n ∈ N.

(1.9)

θ,δ Definition 1.1 The pair (µ0 , Nm ), where µ0 ∈ M(H0 ) and Nθ,δ m is defined by (1.8), is called a generalized hierarchical model.

With each µn one may associate a random element X (n) by setting Prob(X (n) ∈ B) = µn (B),

B ∈ B0 ,

n ∈ N0 .

For θ = 0, they sequence satisfy the recursive relation ³

(n−1)

X (n) = m−(1+δ)/2 X1

´

(n−1) + . . . + Xm ,

(1.10)

(n−1)

where Xk , k = 1, . . . m are independent identical copies of X (n−1) . Iterating (1.10) one obtains ³

(0)

(0)

X (n) = m−(1+δ)n/2 X1 + . . . + Xmn

´

thus, the sequence {µn }n∈N0 defined by (1.9) will be asymptotically degenerate at zero for θ = 0 and arbitrary δ > 0. By this we means that this sequence 3

converges to the δ−measure concentrated at zero. One may expect that such a convergence will hold also for small positive values of θ,which would mean (n) the weak dependence of Xk , k = 1, 2 . . . . In this situation it is quite natural to ask whether does exist a threshold value of θ, say θ∗ , which bounds the region of weak dependence and such that, for θ = θ∗ and δ > 0, the corresponding sequence of measures is no longer asymptotically degenerate. The (n) latter fact would mean the appearance of the strong dependence between Xk , k = 1, 2, . . . at θ = θ∗ . A similar phenomenon in the models of equilibrium statistical mechanics is known as a critical point behavior [1] - [3]. To study the asymptotic properties of the above introduced sequence of measures µn one may use the Laplace transformation Z

def

ϕµ (x) =

exp ((x, ξ)0 ) µ(dξ),

x ∈ H0 ,

(1.11)

H0

which, for a measure µ satisfying the condition (1.7), defines the function ϕµ : H0 → R. To study {ϕµn } one has to construct the map ϕµn−1 7→ ϕµn . The Laplace transform of the convolution of measures is the product of their Laplace transforms. To represent the factor exp (. . .) on the right-hand side of (1.8) one has to take a measure on a wider Hilbert space H, which would contain H0 as a dense subset. We suppose that the embedding O : H0 → H is a positive Hilbert-Schmidt operator. Then its canonical representation is O=

X√

sk (·, ξk )0 ηk ,

k∈N

X

sk < ∞;

sk > 0,

∀k ∈ N,

(1.12)

k∈N

where the orthonormal families {ξk }k∈N , {ηk }k∈N are complete, i.e. are the bases of H0 and H respectively. Let O∗ : H → H0 stand for the adjoint operator and def

S = OO∗ =

X

sk (·, ηk )ηk ,

S : H → H.

(1.13)

k∈N

As a strictly positive operator, O is invertible and its inverse O−1 = J may be defined on the set H0 ⊂ H. It is an unbounded operator and maps its domain onto the Hilbert space H0 . Let also J ∗ stand for the adjoint operator, which acts from a dense subset D(J ∗ ) of the space H0 into H. Obviously, J ∗ = (O∗ )−1 . More details on the mappings of this kind acting between the spaces H, H0 can be found in the first chapter of [4]. 4

Let γS be the Gaussian measure on H for which the above introduced S is the covariance operator. For x ∈ D(J ∗ ), one may write Ã

!

Z n√ o θ 2 exp |x|0 = exp θ(J ∗ x, y) γS (dy). 2

(1.14)

H

This representation may be extended to H0 since, for every x ∈ H0 , the map (J ∗ x, ·) = (x, J·)0 : H → R is a measurable linear functional on H. Given a measure µ on H0 , we introduce def

fµ (y) = ϕµ (Jy),

y ∈ H0 ⊂ H.

(1.15)

Now let us assume that the initial element of the sequence of measures (1.9) is such that the function fµ0 may be extended to an appropriate function on H. Then by (1.15), (1.11) and (1.8), one has in (1.9) n o √ 1 Z Z fµn (y) = exp σ(Jy, ξ)0 + σ θ(J ∗ ξ, η) γS (dη)µ?m n−1 (dξ) Zn H H0

=

1 Zn

Z H

  



Z

exp

n³

√

J(σy + σ θη), ξ

´ o 0

 µ?m n−1 (dξ) γS (dη)

H0

³ √ ´im 1 Z h fµn−1 σy + σ θη γS (dη), = Zn

(1.16)

H

where σ = m−(1+δ)/2 and Zn is defined by the condition fµn (0) = 1. Then the generalized hierarchical model (µ0 , Nθ,δ m ) may be realized as a sequence {fn }n∈N0 , generated by the transformation (1.3) with σ = m−(1+δ)/2 .

2

Spaces of Holomorphic Functions

The domain and range spaces of Gσ,θ m,S are chosen as spaces of holomorphic functions defined on the complexifications of H and H0 . Given a real separable Hilbert space X , the complexification X c stands for the following direct sum def X c = X ⊕iX . Equipped with the norm |·|X c such that |ξ +iη|2X c = |ξ|2X +|η|2X it turns into a separable complex Banach space. Definition 2.1 A function f : U → C is said to be G-holomorphic on U, if for any u ∈ U, v ∈ X c , the map C 3 λ 7→ f (u + λv) is holomorphic, as a function of a single complex variable, on some neighborhood of 0 ∈ C. A G-holomorphic function is called holomorphic if it is continuous on U. 5

The set of holomorphic on X c functions will be denoted by Hol(X c ). Let B = {z ∈ X c | |z| ≤ 1}. For an appropriate f ∈ Hol(X c ) and r > 0, we set def

φf (r) = kf krB = sup {|f (z)|}.

(2.1)

z∈rB

Further, f ∈ Hol(X c ) is a holomorphic function of bounded type if, for every r ≥ 0, kf krB < ∞.

(2.2)

We consider even holomorphic functions on X c , for which one may f (z) =

∞ X

1 d d2k f (0)(z), (2k)! k=0

(2.3)

2k f (0) is a continuous in z ∈ X c , 2k-linear homogeneous polynomial where dd taking values in C. For positive β and an even function f , we set

³

def

´

kf k1,β = sup {|f (z)| exp −β|z|2 } = sup{φf (r) exp(−βr2 )}, z∈X c

(2.4)

r≥0

( def

kf k2,β = sup β

)

−k

k∈N0

k! 2k f (0)k kdd B . (2k)!

(2.5)

Proposition 2.1 Given an even function f ∈ Hol(X c ) and a certain γ > 0, let the norm kf k1,γ be finite. Then for every ² > 0, its norms kf k1,γ+² and kf k2,γ+² are also finite and the following estimates hold kf k1,γ+² ≤ kf k2,γ+² ≤ K(γ, ²)kf k1,γ ,

(2.6)

where the constant K(γ, ²) is independent of f . For β > 0, we define def

Bβ (X ) = {f ∈ Hol(X c ) | f (x) = f (−x),

kf k2,β < ∞},

(2.7)

and equip this set with the k · k2,β -norm topology. Further, for α ≥ 0, let def

Aα (X ) =

\

Bβ (X ).

(2.8)

β>α

6

The latter space equipped with the topology Tα generated by the family of norms {k · k2,β | β > α} will play a key role in our study. Theorem 2.1 For any β > 0, (Bβ (X ), k · k2,β ) is a Banach space; for every α ≥ 0, (Aα (X ), Tα ) is a Fr´echet space. Let T bnd stand for the topology of uniform convergence on bounded subsets of X c . Theorem 2.2 For any bounded subset of Aα (X ), the topologies induced on it by T bnd and Tα coincide.

3

The Transformation

+ Given β > 0 (resp. α ≥ 0), let B+ β (X ) (resp. Aα (X )) stand for the subset of Bβ (X ) (resp. Aα (X )) consisting of the functions which are positive on X ⊕ i{0}.

For every mapping T : Aα (X ) → Aγ (X ), one may find β > α and δ > γ such that Bβ (X ) 3 f 7→ T (f ) ∈ Bδ (X ). In what follows, such a mapping may be defined also on the corresponding Banach spaces. A mapping T : Aα (X ) → Aγ (X ) will be said to be Fr´echet-differentiable on Aα (X ) = ∩β>α Bβ (X ) if the corresponding mappings acting between the Banach spaces Bβ (X ), β > α and Bδ (X ), δ > γ are Fr´echet differentiable in the usual sense. Now let H be the Hilbert space which has appeared in the first section. For appropriate f ∈ A+ α (H), we set Gσ,θ m,S (f )(y)

=

Z h

√ im f (σy + σ θη) γS (dη),

y ∈ H,

(3.1)

H

where σ > 0, θ ≥ 0 and m ∈ N \ {1} are the parameters of this operator. We recall that S : H → H is a strictly positive trace class operator. For further convenience, the operator norm kSk = s1 will √ be put to be equal one. In the sequel we restrict σ to be in the interval (0, 1/ m). Set def

αmax =

1 − mσ 2 . 2mσ 2 θ

(3.2)

Theorem 3.1 For any m ∈ N \ {1} and every α ≤ αmax , the transformation Gσ,θ m,S , defined by (3.1), is a continuous self-mapping of Aα (H). Moreover, it echet-differentable on Aα (X ), its Fr´echet maps A+ α (H) into itself and is Fr´ derivative is 7

½³

´0 ¾

Gσ,θ m,S (f ) g (y) =m

(3.3)

Z h

√ im−1 √ f (σy + σ θη) g(σy + σ θη)γS (dη).

H

Set Q = {Q ∈ N | |Q| < ∞}, where |Q| stands for cardinality. For a nonempty Q, let HQ be the linear span of the eigenvectors {ηs }s∈Q of the operator S. Let also WQ be the orthogonal projector onto HQ from H. For Q ∈ Q, we set AQ = 2αmax S −1 WQ ,

A∅ = 0.

(3.4)

Theorem 3.2 Each an element of the family µ

2

|Q|/2(m−1)

A = {fQ (y) = [σ m]

¶

1 exp (AQ y, y) 2

| Q ∈ Q} ⊂ Aαmax (H),

is a fixed point of Gσ,θ m,S . Let P stand for the set of all complex valued continuous polynomials on H. Given Q ∈ Q, we set Qc = N \ Q and PQ = {p ∈ P | ∀x ∈ H : PQc = {p ∈ P | ∀x ∈ H :

p(x) = p(WQ x)}; p(x) = p((1 − WQ )x)}.

(3.5)

To study the stability of the fixed points from A we shall solve the eigenvalue problems ´0

³

Gσ,θ m,S (fQ ) g = λg,

g ∈ Aαmax (H).

(3.6)

Theorem 3.3 For every nonempty Q and for any n ∈ N0 , there exists pn ∈ PQ such that the function g = fQ pn ∈ Aαmax (H) solves (3.6) with λn = m1−2n σ −2n . For every Q and for any n ∈ N0 , there exists a polynomial pcn ∈ PQc such that the function g = fQ pcn ∈ Aαmax (H) solves (3.6) with λcn = mσ 2n . For these polynomials, deg(pn ) = deg(pcn ) = n. The proof of the above theorems is based on the following assertions. Proposition 3.1 Let S and γS be as above and let A be a positive symmetric bounded linear operator on H, such that SA < 1.

(3.7)

Then the expression µ

¶

1 exp (Ax, x) γS (dx) = KB γB (dx) 2 8

(3.8)

defines on H a symmetric Gaussian measure γB with the covariance operator B = (1 − SA)−1 S.

(3.9)

Here Z

KB = H

½

¾

1 1 exp (Ax, x) γS (dx) = [det(1 − SA)]− 2 . 2

(3.10)

Lemma 3.1 Given positive σ and θ, let a symmetric positive bounded operator A˜ : H → H obey the condition S A˜ < 1/(σ 2 θ). Then for an appropriate function g˜ : H → R, the following is true (

Z

)

³ √ √ ´ √ ´ σ2 ³ ˜ exp A(y + θη), y + θη g˜ σ[y + θη] γS (dη) = 2

H

)

(

2 ³ ´ ˜ − σ 2 θS A) ˜ −1 y, y × ˜ exp σ A(1 = K(S, A) 2 Z ³ h √ i´ ˜ −1 y + θη γB (dη), × g˜ σ (1 − σ 2 θS A)

(3.11)

H

where ˜ −1 S, B = (1 − σ 2 θS A)

h

i−1/2

˜ = det(1 − σ 2 θS A) ˜ K(S, A)

.

(3.12)

Corollary 3.1 Let S, Gσ,θ m,S and A be as above, let also h

SA < mσ 2 θ

i−1

.

Then for the function ½

f (y) = exp

¾

1 (Ay, y) g(y), 2

g ∈ A0 (H),

one has ¾

½

´ m 2³ σ A(1 − mσ 2 θSA)−1 y, y × 2 ³ ´ σ,θ × Gm,C (g) (1 − mσ 2 θSA)−1 y ,

Gσ,θ m,S (f )(y) = K(S, mA) exp

9

(3.13)

where C = (1 − mσ 2 θSA)−1 S.

4

(3.14)

Extension to Laplace Transforms

As it was demonstrated above, Gσ,θ m,S may be employed to describe generalized hierarchical models (see Definition 1.1) if the initial measure µ0 is such that the function ϕµ0 (Jy), y ∈ H0 ⊂ H may be extended to an element of the σ,θ domain of Gσ,θ m,S , established by Theorem 3.1. Now our aim is to extend Gm,S to the functions which are not necessarily extendable to functions holomorphic on the whole Hc but are the Laplace transforms of measures from the family M(H0 ). For any µ ∈ M(H0 ), the Laplace transform ϕµ , given by (1.11) exists for any x ∈ H0 . Theorem 4.1 For every µ ∈ M(H0 ), the Laplace transform ϕµ may be extended to a holomorphic on H0c function of bounded type. For x = xr , ϕµ (x) > 0 one may prove µ

0 < ϕµ (x) ≤ Aµ (a) exp

¶

1 2 |x| , 4a 0

x ∈ H0 .

(4.1)

Let us introduce Ma (H0 ) = {µ ∈ M(H0 ) | Aµ (a) < ∞}, a > 0, Lα (H0 ) = {ϕ ∈ Hol(H0c ) | ∃ µ ∈ M1/4α (H0 ) : ϕ = ϕµ }.

(4.2)

One may show that Lα (H0 ) ⊂ Aα (H0 ). For an appropriate function ϕ, we set Hσ,θ m,S (ϕ)(x)

=

Z h

im √ ϕ(σx + σ θJy) γS (dy),

x ∈ H0 ,

(4.3)

H

where m, S, σ, θ are the same as above. Theorem 4.2 For any m ∈ N \ {1} and α ≤ αmax given by (3.2), the transformation Hσ,θ m,S may be applied to ϕ ∈ Lα (H0 ) and the result, up to a positive multiplier, will be a function from Lα (H0 ), i.e. ³

´

σ,θ Hσ,θ m,S (ϕ)(x)/Hm,S (ϕ)(0) ∈ Lα (H0 ).

10

(4.4)

If the function ϕ ∈ Lα (H0 ) is such that the function f (y) = ϕ(Jy), y ∈ H0 ⊂ H may be extended to an element of Aα (H), then, for these ϕ and f , one has σ,θ Hσ,θ m,S (ϕ)(Jy) = Gm,S (f )(y).

(4.5)

To prove this theorem we will use the following lemma. Lemma 4.1 Given µ1 , µ2 , . . . , µm ∈ M(H0 ), let a1 , a2 , . . . , am ∈ (0, +∞) be such that Aj (aj ) < ∞, j = 1, 2, . . . , m. Then for b > 0 obeying the condition −1

b

=

m X

a−1 j

(4.6)

j=1

the following estimate holds Aµ1 ?µ2 ?...?µm (b) ≤ Aµ1 (a1 )Aµ2 (a2 ) . . . Aµm (am ).

(4.7)

References [1] S. Albeverio, Yu. Kondratiev, Yu. Kozitsky, Critical properties of a quantum hierarchical model, Lett. Math. Phys., 40 (1997), 287-291. [2] S. Albeverio, Yu. Kondratiev, Yu. Kozitsky, Absence of critical points for a class of quantum hierarchical models, Comm. Math. Phys., 187 (1997), 1-18. [3] S. Albeverio, Yu. Kondratiev, A. Kozak, Yu. Kozitsky, A system of quantum anharmonic oscillators with hierarchical stucture: critical point convergence, BiBoS preprint, Universit¨at Bielefeld, Bielefeld, 2002. [4] Yu.M. Berezansky, Yu.G. Kondratiev, Spectral Methods in Infinite Dimensional Analysis, Kluwer Academic Publishers, Holland (1994). [5] Yu.L.Daletskii, Infinite-dimensional elliptic operators and parabolic equations connected with them, Russian Math. Survey 22 (1967), 1–53. [6] D.A. Dawson, A. Greven, Hierarchical model of interacting diffusions: multiple time scale phenomena, phase transitions and pattern of cluster-formation, Probab. Theory Related Fields 96 (1993), 435-473. [7] A. Kozak, Yu.Kozitski, A nonlinear dynamical system as a solution of a FDE in Hilbert space, Nonlinear Anal. Ser. A: Theory Methods, 47 (2001), 3949-3961. [8] Yu. Kozitsky, L. Wolowski, Laguerre entire functions and related locally convex spaces, Comlex Variables, 44 (2001), 225-244. [9] Yu. Kozitsky, L. Wolowski, A nonlinear dynamical system on the set of Laguerre entire functions, Nonlinear Anal. Ser. A: Theory Methods, 49 (2002), 61-86.

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