## 1 Introduction

The studying of a random processes in infinite dimension Banach spaces and its description by a partial differential equation for a functions on the Banach space are the important topics of contemporary mathematics (see [4, 8, 9]). To the investigation of the above topics and to construct the quantum theory of infinite dimension Hamiltonian systems the analogs of the Lebesgue measure on the infinite dimension linear space are introduced in the works [1, 10, 13, 17].

To study the random walks in the real Hilbert space *E* we introduce a class of measures on the Hilbert space which are invariant with respect to a shift on an arbitrary vector of the space *E* (see [10, 14]). For any choice of such measure we construct the Hilbert space *ℋ* of complex-valued functions any of each is square integrable with respect to this measure. We study operators of argument shifts on the spaces *ℋ*.

We study the random shift operator on the vector whose distribution on Hilbert space *E* is given by a semi-group *γ _{t}*,

*t*≥ 0, of Gaussian measures with respect to the convolution. We prove that the mean values of the random shift operator form the one-parametric semigroup

**U**(

*t*),

*t*≥ 0, of self-adjoint contractions in the space

*ℋ*. The criteria of strong continuity of this semigroup

**U**is obtained.

We prove that if the semigroup **U** is strongly continuous in the space *ℋ* then for any *t* > 0 the image **U**(*t*) *f* of any vector *f* ∈ *ℋ* has the derivatives of any order in the direction of any eigenvector of covariation operator **D** of Gaussian measure *γ*_{1}. Therefore the space of smooth functions is defined as the image of the space *ℋ* under the actions of the operators **U**(*t*), *t* > 0, of semigroup **U**.

**D**in the space

*E*the Sobolev space

*u*∈

*ℋ*such that (∂

*)*

_{k}*∈*

^{l}u*ℋ*for any

*l*∈ {1,...,

*m*} and any

*k*∈ ℕ; and the following series converges

Here {*d _{k}*} is the sequence of eigenvalues of the operator

**D**and {

*e*} is the sequence of corresponding eigenvectors. The function

_{k}*u*∈

*ℋ*has the derivative ∂

*∈*

_{h}u*ℋ*in the direction of the unite vector

*h*∈

*E*if the following equality

We study the connection of the random walks in the space *E* with the self-adjoint analogue of Laplace operator whose domain is the Sobolev space. We prove that the analogue of Laplace operator is the generator of the semigroup of self-adjoint operators arising as the mean value of random shift operators. The properties of smooth function space embedding into the Sobolev spaces are studied. The analogue of Gauss theorem is obtained for the simplest domains such as the rectangle in the space *E*. The correctness of the problem for Poisson equation in the rectangle with homogeneous Dirichlet condition is obtained and the variational approach of the solving of this problem is constructed.

## 2 A class of shift invariant measures on a Hilbert space

According to A. Weil theorem there is no Lebesgue measure on the infinite dimensional separable normed real linear space *E*, i.e. there is no Borel *σ*-additive *σ*-finite locally finite measure on the space *E* which is translation-invariant. Therefore an analogue of the Lebesgue measure is defined as an additive function on some ring of subsets of the space *E* which is translation-invariant. In this paper we present the analogue of the Lebesgue measure which is *σ*-finite and locally finite but not Borel and not *σ*-additive measure (see [10, 11, 12]). In the papers [1, 16, 17] the analogue of the Lebesgue measure is considered as the measure which is Borel and *σ*-additive but not *σ*-finite and not locally finite.

We study invariant measures on a real separable Hilbert space *E*, which are invariant with respect to any shift. In this article finite-additive analogues of the Lebesgue measure are constructed. The non-negative finite-additive translation-invariant measure *λ* is defined on the special ring *ℛ* of subsets from a space *E* in the work [10]. The ring *ℛ* contains all infinite-dimensional rectangles whose products of side lengths are absolutely convergent.

Now we describe some class of translation-invariant measures on separable Hilbert space *E* any of each is the restriction of measure *λ* from the work [10] on a ring *ℛ _{ℰ}* depending on the chois of an orthonormal basis

*ℰ*= {

*e*} in te space

_{j}*E*. Let

*𝒮*be a set of orthonormal bases in the space

*E*. Firstly we describe a class of measures on the space

*E*which are invariant with respect to the shift on any vector of this space.

*E*is the set Π ⊂

*E*such that there is an orthonormal basis {

*e*} ≡

_{j}*ℰ*in

*E*and there is an elements

*a,b*∈

*l*

_{∞}such that

The rectangle (1) is noted by the symbol Π* _{ℰ,a,b}*.

*λ*(Π) = 0 if Π = ∅, and let

*.*

_{ℰ,a,b}For any orthonormal basis *ℱ* = {*f _{k}*} of the space

*E*the symbol

*K*(

*ℱ*) note the set of measurable rectangles with the edges collinear to the vectors of ONB

*ℱ*. Let the symbol

*r*notes the minimal ring of subsets containing the set of measurable rectangles

_{ℱ}*K*(

*ℱ*).

*[11] For any orthonormal basis ℱ* = {*f _{k}*}

*of the space E there exists the unique measure λ*:

_{ℱ}*r*→ [0, +∞)

_{ℱ}*such that the equality (3) holds for any rectangle*Π

*∈*

_{ℱ,a,b}*𝒦*

_{ℱ}. The measure λ_{ℱ}has the unique completion onto the ring ℛ_{ℱ}which is completion of the ring r_{ℱ}by measure λ_{ℱ}.**Note 2.** Here the ring *ℛ _{ℱ}* consists on the sets

*A*⊂

*E*such that

*A*with respect to the measure

*λ*.

_{ℱ}**Note 3.** Note that there are translation-invariant measures on the space *E* of another type which is countable additive but not *σ*-finite (see [17]). There are measures on infinite dimensional topological vector spaces which are translation-invariant with respect to only some subspace of acceptable vectors (see [14]).

### 2.1 Quadratically integrable functions

Now we define space of quadratically integrable functions with respect to *λ _{ℰ}*. Since we will use it very often, we define it concisely

*ℋ*=

_{ℰ}*L*

_{2}(

*E*,

*ℛ*,

_{ℰ}*λ*ℂ).

_{ℰ},*𝒮*(

*E*,

*ℛ*, ℂ) be the linear space hull over field ℂ of indicator functions of the sets from the ring

_{ℰ}*ℛ*. Let

_{ℰ}*β*be the sesquilinear form on the space

_{ℰ}*𝒮*(

*E*,

*ℛ*, ℂ) which is defined by the following conditions:

_{ℰ}*A*,

*B*∈

*𝒦*; for any functions

_{ℰ}*u*,

*v*∈

*𝒮*(

*E*,

*ℛ*, ℂ) such that

_{ℰ}*β*(

_{ℰ}*u*,

*v*) is given by the equality

This sesquilinear form on the space *𝒮* (*E*, *ℛ _{ℰ}*, ℂ) is Hermitian and nonnegative.

The function *u* ∈ *𝒮* (*E*, *ℛ _{ℰ}*, ℂ) is called equivalent to the function

*u*∈

*𝒮*(

*E*,

*ℛ*, ℂ) iff

_{ℰ}*β*(

_{ℰ}*u*−

*v*,

*u*−

*v*) = 0. The linear space

*𝒮*

_{2}(

*E*,

*ℛ*,

_{ℰ}*λ*ℂ) of the equivalence classes of functions of the space

_{ℰ},*𝒮*(

*E*,

*ℛ*, ℂ) endowing with the sesquilnear form

_{ℰ}*β*is the pre-Hilbert space. The Hilbert space

_{ℰ}*L*

_{2}(

*E*,

*ℛ*,

_{ℰ}*λ*ℂ) ≡

_{ℰ},*ℋ*is defined as the completion of the space

_{ℰ}*𝒮*

_{2}(

*E*,

*ℛ*,

_{ℰ}*λ*, ℂ).

_{ℰ}Thus for any ONB *ℰ* in the space *E* there are the ring of subsets *ℛ _{ℰ}* of

*λ*-measurable sets, the measure

_{ℰ}*λ*:

_{ℰ}*ℛ*→ [0, +∞) and the Hilbert space

_{ℰ}*ℋ*of complex valued

_{ℰ}*λ*-measurable square integrable functions on the space

_{ℰ}*E*. Since the pre-Hilbert space

*𝒮*

_{2}(

*E*,

*ℛ*,

_{ℰ}*λ*, ℂ) of a simple functions is dense linear manifold in the space

_{ℰ}*ℋ*then the Hibert space

_{ℰ}*ℋ*is the space of continuous linear functionals on the pre-Hilbert space

_{ℰ}*𝒮*

_{2}(

*E*,

*ℛ*,

_{ℰ}*λ*, ℂ).

_{ℰ}*[10] The space ℋ _{ℰ} is not separable.*

### 2.2 The products of the spaces with finite additive measures

Let *ℰ* = {*e*_{1}, *e*_{2}, ...} be an ONB in the space *E*. Let *E*_{1} be the Hilbert space with the ONB *ℰ*_{1} = {*e*_{2}, *e*_{3}, ...}. Then *E* = ℝ ⊕ *E*_{1}, *E* ∋ *x* = (*x*_{1}, *ξ*) ∈ ℝ ⊕ *E*_{1}, where *x*_{1} ∈ ℝ; *ξ* ∈ *E*_{1}.

Let *𝒥* be the isomorphism of the Hilbert space *E* onto the Hilbert space *E*_{1} such that *ℰ*_{1} = *𝒥* (*ℰ*). Let *λ _{ℰ}* be a complete translation invariant measure on the space

*E*such that the measure

*λ*is defined on the ring

_{ℰ}*ℛ*by the theorem 1. Let

_{ℰ}*ℛ*

_{ℰ1}=

*𝒥*(

*ℛ*) and

_{ℰ}*λ*

_{ℰ1}is the measure on the space (

*E*

_{1},

*ℛ*

_{ℰ1}) such that

*λ*

_{ℰ1}(

*A*) =

*λ*(

_{ℰ}*𝒥*

^{−1}(

*A*)) ∀

*A*∈

*ℛ*

_{ℰ1}.

Let **l**_{ℝ} be the Jordan measure on the real line ℝ. Let *r*( ℝ) be a ring of measurable by Jordan subsets of real line ℝ. Remind that *K _{ℰ}* and

*K*

_{ℰ1}are the collections of measurable rectangles in the spaces

*E*and

*E*

_{1}whose edges are collinear to the vectors of ONB

*ℰ*and

*ℰ*

_{1}respectively;

*r*and

_{ℰ}*r*

_{ℰ1}are the minimal rings containing the collections of sets

*K*and

_{ℰ}*K*

_{ℰ1}respectively.

We will use the following notations Π = Π′ × Π″ ⊂ ℝ × *E*_{1} where Π′ the finite segment of real line and Π, Π″ are the measurable rectangles in the spaces *E*, *E*_{1} respectively.

*([2], lemma 3.3) The inner measure of the set X*⊂

*E is defined by the equality*

*where supremum is defind over the set of finite union of measurable rectangles but not on the hole ring r*

_{ℰ}.*([2], lemma 3.4). Let A* = *g* × Π, *where* Π ∈ *𝒦*_{ℰ1}*and λ*_{ℰ1}(Π) ≠ 0. *Then A* ∈ *ℛ _{ℰ} iff g* ∈

*r*(ℝ).

*In this case the following equality λ*(

_{ℰ}*A*) =

**l**

_{ℝ}(

*g*)

*λ*

_{ℰ1}(Π)

*holds.*

The collection *𝒦 _{ℰ}* of measurable rectangles is the part of the following collection of sets {

*A*

_{0}×

*A*

_{1},

*A*

_{0}∈

*r*( ℝ),

*A*

_{1}∈

*ℛ*

_{ℰ1}}; the last collection of sets is the part of the ring

*ℛ*. Since the ring

_{ℰ}*r*is the minimal ring containing the collection of sets

_{ℰ}*𝒦*and the ring

_{ℰ}*ℛ*is the completion of the ring

_{ℰ}*r*by the measure

_{ℰ}*λ*, then the ring

_{ℰ}*ℛ*is the comletion by measure

_{ℰ}*λ*of the minimal ring, containing the collection of the sets {

_{ℰ}*A*

_{0}×

*A*

_{1},

*A*

_{0}∈

*r*( ℝ),

*A*

_{1}∈

*ℛ*

_{ℰ1}}. Hence the following statement holds.

*The space with finite additive measure* (*E*, *r _{ℰ}*,

*λ*)

_{ℰ}*is the prodict of the spaces with finite additive measures*( ℝ,

*r*( ℝ),

**l**

_{ℝ})

*and*(

*E*

_{1},

*r*

_{ℰ1},

*λ*

_{ℰ1})

According to the lemma 1 [5] (page 222) the space with finite additive measure (*E*, *r _{ℰ}*,

*λ*) is the product of the spaces with the finite additive measures ( ℝ,

_{ℰ}*r*( ℝ),

**l**

_{ℝ}) and (

*E*

_{1},

*r*

_{ℰ1},

*λ*

_{ℰ1}). In fact, since the rings

*ℛ*and

_{ℰ}*ℛ*

_{ℰ1}is obtained by using of completions by measures

*λ*and

_{ℰ}*λ*

_{ℰ1}procedure from the rings

*r*and

_{ℰ}*r*

_{ℰ1}respectively then the measure

*λ*be a unique finite additive measure which is defined on the ring

_{ℰ}*ℛ*and satisfies the conditions

_{ℰ}*λ*(

_{ℰ}*A*

_{0}×

*A*

_{1}) =

**l**

_{ℝ}(

*A*

_{0})

*λ*

_{ℰ1}(

*A*

_{1}) ∀

*A*

_{0}∈

*r*( ℝ),

*A*

_{1}∈

*ℛ*

_{ℰ1}.

A tensor product of the finite additive measures *μ* = *μ*_{1} ⊗ *μ*_{2} on the space *X* = *X*_{1} × *X*_{2} is the measure *μ* on the space *X* which the completion of the measure *μ*_{1} × *μ*_{2}. Here *μ*_{1} × *μ*_{2} is the measure which satisfies following two conditions:

- 1)it is defined on the minimal ring containing the collection of sets {
*A*_{1}×*A*_{2},*A*_{1}∈*R*_{1},*A*_{2}∈*R*_{2}}, - 2)it satisfies the equality
*μ*_{1}×*μ*_{2}(*A*_{1}×*A*_{2}) =*μ*_{1}(*A*_{1})*μ*_{2}(*A*_{2}) ∀*A*_{1}∈*R*_{1},*A*_{2}∈*R*_{2}.

*The following equality λ _{ℰ}* =

**l**

_{ℝ}⊗

*λ*

_{ℰ1}

*holds in the sense of definition 1.*

In fact, the procedures of definition of the measure *λ _{ℰ}* and the measure

**l**

_{ℝ}⊗

*λ*

_{ℰ1}have the following common constructions.

The definition the measure *λ _{ℰ}* consists of three parts:

- The function of a set is defined firstly on the collection of measurable rectangles
*𝒦*;_{ℰ} - the function
*𝒦*→ ℝ is extended onto the measure_{ℰ}*λ*on the minimal ring*r*containing the collection of sets_{ℰ}*𝒦*;_{ℰ} - the measure
*λ*:*r*→ ℝ is extended onto the ring_{ℰ}*ℛ*which is completion of the ring_{ℰ}*r*by measure_{ℰ}*λ*.

In the case of the measure **l**_{ℝ} ⊗ *λ*_{ℰ1} the collection of a sets {*A*_{1} × *A*_{2} | *A*_{1} ∈ *r*( ℝ), *A*_{2} ∈ *𝒦*_{ℰ1}} is used instead of the collection *𝒦 _{ℰ}* on the first step. According to definition the equality

**l**

_{ℝ}⊗

*λ*

_{ℰ1}(

*I*

_{1}× Π

_{1}) =

*λ*(

_{ℰ}*I*

_{1}× Π

_{1}) holds for any measurable rectangle

*I*

_{1}× Π

_{1}where

*I*

_{1}∈

*r*( ℝ), Π

_{1}∈

*𝒦*

_{ℰ1}. Therefore the inequalities

*A*⊂

*E*. Hence if a set

*A*⊂

*E*is measurable with respect to the measure

*λ*then it is measurable with respect measure

_{ℰ}**l**

_{ℝ}⊗

*λ*

_{ℰ1}and the the extensions of the measures

*λ*and

_{ℰ}**l**

_{ℝ}⊗

*λ*

_{ℰ1}coincides on the ring

*ℛ*.

_{ℰ}On the other hand any set of type {*A*_{1} × *A*_{2} | *A*_{1} ∈ *r*( ℝ), *A*_{2} ∈ *𝒦*_{ℰ1}} belongs to the ring *ℛ _{ℰ}* and in this case the equality

*λ*(

_{ℰ}*A*

_{1}×

*A*

_{2}) = (

**l**

_{ℝ}⊗

*λ*

_{ℰ1})(

*A*

_{1}×

*A*

_{2}) holds. Since the measure

*λ*:

_{ℰ}*ℛ*→ ℝ is complete then the continuation of the function of a set (

_{ℰ}**l**

_{ℝ}⊗

*λ*

_{ℰ1}) : {

*A*

_{1}×

*A*

_{2}|

*A*

_{1}∈

*r*( ℝ),

*A*

_{2}∈

*𝒦*

_{ℰ1}} → ℝ by means of continuation on the minimal ring (steps 2)) and completion (step 3)) coincides with the measure

*λ*.

_{ℰ}*The linear space* span({*χ*_{Π}, Π ∈ *𝒦 _{ℰ}* })

*is dense in the Hilbert space L*

_{2}(

*E*,

*ℛ*,

_{ℰ}*λ*, ℂ).

_{ℰ}In fact, according to definition the Hilbert space *L*_{2}(*E*, *ℛ _{ℰ}*,

*λ*, ℂ) is the closure of the linear space

_{ℰ}*L*

_{2}(

*E*,

*ℛ*,

_{ℰ}*λ*ℂ). Since the ring

_{ℰ},*ℛ*is the completion of the ring

_{ℰ}*r*by the measure

_{ℰ}*λ*then the following equality

_{ℰ}*A*∈

*r*is the finite union of disjoint sets

_{ℰ}*B*is the complement of a measurable rectangle to finite union of measurable rectangles:

_{k}*χ*∈ span({

_{A}*χ*

_{Π}, Π ∈

*𝒦*}) for any

_{ℰ}*A*∈

*r*. Consequently,

_{ℰ}*Morphism ℐ mapping element of L*_{2}( ℝ) ⊗ *L*_{2}(*E*_{1}, *ℛ*_{ℰ1}, *λ*_{ℰ1},ℂ), *which is limit of fundamental sequence f _{k}* ⊗

*v*⊗

_{k}into limit of sequence f_{k}*v*

_{k}in space L_{2}(

*E*,

*ℛ*,

_{ℰ}*λ*, ℂ)

_{ℰ}*provides us with canonical isomorphism between these two space.*

Space *L*_{2}(*E*, *ℛ _{ℰ}*,

*λ*, ℂ) according to lemma 9 is a completion of span({

_{ℰ}*χ*,

_{A}*A*∈

*𝒦*}) with respect to norm ‖ · ‖

_{ℰ}_{L2(E,ℛℰ, λℰ},

_{ℂ}), defined by sesquilinear form

*β*, see (1). So,

_{ℰ}*L*

_{2}(

*E*,

*ℛ*,

_{ℰ}*λ*, ℂ) is completion of the space span({

_{ℰ}*χ*

_{A0×A1},

*A*

_{0}∈

*𝒦*

_{ℝ},

*A*

_{1}∈

*𝒦*

_{ℰ1}}) with respect to norm ‖ · ‖

_{L2 (E,ℛℰ, λℰ},

_{ℂ}), defined by sesquilinear form

*β*. (Here we define by

_{ℰ}*𝒦*

_{ℝ}a set of all bounded intervals of ℝ).

Space *L*_{2}( ℝ) ⊗ *L*_{2}(*E*_{1}, *ℛ*_{ℰ1}, *λ*_{ℰ1}, ℂ) is completion of linear span *ℒ* of elements *f* ⊗ *v*, where *f* ∈ *L*_{2}( ℝ), *v* ∈ *L*_{2}(*E*_{1}, *ℛ*_{ℰ1}, *λ*_{ℰ1}, ℂ) with respect to norm ‖ · ‖_{⊗}, defined by sesquilinear form *β*_{⊗} on *ℒ* with restriction *β*_{⊗} (*f*_{1} ⊗ *v*_{1}, *f*_{2} ⊗ *v*_{2}) = (*f*_{1}, *f*_{2})_{L2( ℝ)}(*v*_{1}, *v*_{2})_{L2}(*E*_{1},*ℛℰ*_{1},*λℰ*_{1},ℂ). Note that in space *L*_{2}( ℝ) linear span *ℒ*_{0} of set of characteristic functions from *𝒦*_{ℝ} is dense linear submanifold, and in space *L*_{2}(*E*_{1}, *ℛℰ*_{1}, *λℰ*_{1}, ℂ) according to lemma 9 linear span *ℒ*_{1} of set of characteristic functions of set from *𝒦ℰ*_{1} is also a dense linear submanifold. That’s why space *L*_{2}( ℝ) ⊗ *L*_{2}(*E*_{1}, *ℛℰ*_{1}, *λℰ*_{1}, ℂ) is exactly a comletion of linear span span{*χA*_{0} ⊗ *χA*_{1}, *A*_{0} ∈ *𝒦*_{ℝ}, *A*_{1} ∈ *𝒦ℰ*_{1}} with respect to norm ‖ · ‖_{⊗}.

*𝒦*

_{ℝ}and any measurable rectangle Π′∈

*𝒦*

_{ℰ1}holds ‖

*χ*

_{Δ}⊗

*χ*

_{Π′}‖

_{⊗}= ‖

*χ*

_{Π}‖

_{L}_{2(}

*,*

_{E,ℛℰ}

_{λℰ},_{ℂ}), where Π = Δ × Π′, then for any sets

*A*

_{0}∈

*𝒦*

_{ℝ},

*A*

_{1}∈

*𝒦*

_{ℰ1}holds

Since linear span span({*χ*_{A0×A1}, *A*_{0} ∈ *𝒦*_{ℝ}, *A*_{1} ∈ *𝒦*_{ℰ1}}) is dense in space *L*_{2}(*E*, *ℛ _{ℰ}*,

*λ*, ℂ), and linear span span({

_{ℰ}*χ*

_{A0}⊗

*χ*

_{A1},

*A*

_{0}∈

*𝒦*

_{ℝ},

*A*

_{1}∈

*𝒦*

_{ℰ1}}) is dense in space

*L*

_{2}( ℝ) ⊗

*L*

_{2}(

*E*

_{1},

*ℛ*

_{ℰ1},

*λ*

_{ℰ1}, ℂ), then (dee equation 2) spaces

*L*

_{2}(

*E*,

*ℛ*,

_{ℰ}*λ*, ℂ) and

_{ℰ}*L*

_{2}( ℝ) ⊗

*L*

_{2}(

*E*

_{1},

*ℛ*

_{ℰ1}

*λ*

_{ℰ1}, ℂ) are isometrically isomorphic and

*L*

_{2}(

*E*,

*ℛ*,

_{ℰ}*λ*, ℂ) =

_{ℰ}*ℐ*(

*L*

_{2}( ℝ) ⊗

*L*

_{2}(

*E*

_{1},

*ℛ*

_{ℰ1}

*λ*

_{ℰ1}, ℂ)).

### 2.3 Partial Fourier transforms

*L*

_{2}( ℝ) is unitary mapping of the space

*L*

_{2}( ℝ) into itself. Therefore the partial Fourier transform

*ℱ*

_{1}with respect to the first coordinate is defined on the space

*L*

_{2}(

*E*,

*ℛ*,

_{ℰ}*λ*, ℂ) =

_{ℰ}*L*

_{2}( ℝ) ⊗

*L*

_{2}(

*E*

_{1},

*ℛ*

_{ℰ1},

*λ*

_{ℰ1},ℂ). The partial Fourier transform

*ℱ*

_{1}is defined on the linear hull of the elements of type

*u*

_{1}(

*x*

_{1})

*u*

_{2}(

*x*),

*u*

_{1}∈

*L*

_{2}( ℝ),

*u*

_{2}∈

*L*

_{2}(

*E*

_{1},

*ℛ*

_{ℰ1},

*λ*

_{ℰ1}, ℂ) by the equality

*û*

_{1},

*∈*

_{k}*L*

_{2}( ℝ) is Fourier transform of the function

*u*

_{1},

*∈*

_{k}*L*

_{2}( ℝ). Since the linear hull of the elements of type

*u*

_{1}(

*x*

_{1})

*u*

_{2}(

*ξ*),

*u*

_{1}∈

*L*

_{2}( ℝ),

*u*

_{2}∈

*L*

_{2}(

*E*

_{1},

*ℛ*

_{ℰ1},

*λ*

_{ℰ1}, ℂ) is dense in the space

*L*

_{2}(

*E*,

*ℛ*,

_{ℰ}*λ*ℂ) then the partial Fourier transform

_{ℰ},*ℱ*

_{1}has the unique continuation up to the unitary transform of the space

*L*

_{2}(

*E*,

*ℛ*,

_{ℰ}*λ*, ℂ) into itself.

_{ℰ}Analogously, partial Fourier transform *ℱ _{n}* with respect to first

*n*coordinates is defined on the space

*L*

_{2}(

*E*,

*ℛ*,

_{ℰ}*λ*, ℂ). According to the properties of Fourier transform of the space

_{ℰ}*L*

_{2}( ℝ

*) with some*

^{n}*n*∈ ℕ; the following statement holds: partial Fourier transform of the space

*L*

_{2}(

*E*,

*ℛ*,

_{ℰ}*λ*, ℂ) with respect to first

_{ℰ}*n*coordinates is unitary mapping of the space

*L*

_{2}(

*E*,

*ℛ*,

_{ℰ}*λ*, ℂ) into itself for any

_{ℰ}*n*∈ ℕ;.

Partial Fourier transform will useful in the studying of the operators of multiplication on coordinate and momentum operator with respect to direction of a vector *e _{j}* of the basis

*ℰ*. It also be used further in the studying of generators of diffusion semigroups and its fraction powers.

## 3 Sobolev spaces and spaces of smooth functions

### 3.1 Averaging of random shifts and space of smooth functions

Let **D** ∈ *B*(*E*) be a nonnegative trace class operator with the orthonormal basis *ℰ* of eigenvectors. Any operator **D** of the above class defines the centered countable additive Gaussian measure *ν*_{D} on the space *E* such that the measure *ν*_{D} has the covariance operator **D** and zero mean value.

*h*∈

*E*is defined on the space

*ℋ*=

_{ℰ}*L*

_{2}(

*E*,

*ℛ*, ℂ) by the equality

_{ℰ}.λ_{ℰ}*h*∈

*H*operator

**S**

*belongs to the Banach space*

_{h}*B*(

*ℋ*) of bounded linear operators in the space

_{ℰ}*ℋ*endowed with the operator norm; moreover

_{ℰ}**S**

*is the unitary operator in the space*

_{h}*ℋ*. Let

_{ℰ}*h*be a random vector of the space

*E*whose distribution is given by the measure

*ν*. Then the mean value

**U**∈

*B*(

*ℋ*) of random shift operator

_{ℰ}**S**

*is given by the Pettis integral*

_{h}According to the paper [12] (see also [15]) the following statement holds.

*Let*

**D**∈

*B*(

*E*)

*be a nonnegative trace class operator with the orthonormal basis ℰ of eigenvectors. Then one-parametric family of operators*

*is a one-parametric semigroup of self-adjoint contractions in the space ℋ*

_{ℰ}. The semigroup**U**

*≥ 0,*

_{t}, t*is strong continuous in the space ℋ*

_{ℰ}if and only if*is trace class operator.*

A function *u′ _{j}* ∈

*ℋ*is called the derivative of the function

_{ℰ}*u*∈

*ℋ*in the direction of a unite vector

_{ℰ}*e*if the following equality holds

_{j}*[see [11], lemmas 7.1, 7.2]. Let*{

*e*} ≡

_{j}*ℰ be the orthonormal basis of eigenvectors of positive trace class operator*

**D**.

*If ν*

_{D}

*be a probability Gaussian measure on the space E with covariance operator*

**D**

*and*

*t*≥ 0,

*u*∈

*ℋ*∈ ℕ;

_{ℰ}then for any l*there is the number c*> 0

_{l}*such that for any u*∈

*ℋ*∈ ℕ;,

_{ℰ}, j*t*> 0

*there is the derivative of the power l*

*and the following estimates take place*

Let **D** be a positive trace class operator in the space *E*. Let
*𝒰*_{D}(*t*)*u*, *t* > 0, *u* ∈ *ℋ _{ℰ}*}. The linear manifold

### 3.2 Sobolev spaces and embedding theorem

For any *a* > 0 the symbol
*u* of the space *ℋ _{ℰ}* such that the following two condition hold

*j*∈ ℕ; there is the derivative

*e*of operator

_{j}**D**;

The space
*ℋ _{ℰ}* (see [3, 11]).

For any numbers *a* > 0 and any *l* ∈ ℕ; the symbol
*u* of the space *ℋ _{ℰ}* such that the following two conditions hold:

*j*∈ ℕ; there is the

*l*-order derivative

*ℋ*(see [3, 11]).

_{ℰ}*[see [11], lemmas 7.1, 7.2]. Let u* ∈ *ℋ _{ℰ}. Let*

**D**

*be a positive trace class operator in the space E such that*

*is the trace class operator. Then for any t*> 0

*the inclusion*

*holds.*

*Let***D***be a positive trace class operator in Hilbert space E such that***D*** ^{γ} is trace class operator with some γ* > 0.

*Let l*∈ ℕ;.

*If b ≥ lα*+

*γ with some α*∈ [

*γ*, +∞)

*then*

*If, moreover, α ≥* 2*γ then the linear manifold**is dense in the space*

The proof of this theorem is published in the work [3].

### 3.3 The traces of a functions on the codimension 1 hyperplane

Let *ℰ* be an orthonormal basis in the space *E*. Let *e*_{1} ∈ *ℰ* and *ℰ*_{1} = *ℰ* \{*e*_{1}}. Let *E*_{1} = (span(*e*_{1}))^{⊥} and *ℋ*_{ℰ1} = *L*_{2}(*E*_{1}, *ℛ*_{1}, *λ*_{ℰ1}, ℂ). Let *ℛ*( ℝ) be a ring of Lebesgue integrable subsets of the space ℝ with finite Lebesgue measure. Let *λ _{L}* be the Lebesgue measure on the space ℝ.

Then accordind to the theorem 10 *ℋ _{ℰ}* =

*L*

_{2}( ℝ,

*ℛ*( ℝ),

*λ*,

_{L}*ℋ*

_{ℰ1})) where

*L*

_{2}( ℝ,

*ℛ*( ℝ),

*λ*,

_{L}*ℋ*

_{ℰ1})) is the space of integrable in Bokhner sense with respect to Lebesgue measure

*λ*mappings ℝ →

_{L}*ℋ*

_{ℰ1}.

*û*=

*ℱ*

_{1}(

*u*). Hence the space

*If*

*then the equivalence class u contains the continuous function ũ*∈

*C*( ℝ,

*ℋ*

_{ℰ1}).

*Moreover, there is the constant C*> 0

*such that*

In the case of separable space *ℋ*_{ℰ1} the proof of this theorem is given in the monograph [6]. In the case under consideration the space *ℋ*_{ℰ1} is not separable. But according to the condition
*ℋ*_{ℰ1} containing the values of the mapping *u* : ℝ → *ℋ*_{ℰ1}. Therefore the proof of the theorem 3.1 by [6] can be applied to the obtaining of the statement of theorem 15.

If
*u* can be considered as the function of the space
*ũ* : ℝ → *ℋ*_{ℰ1} according to theorem 15. Hence we can use the following definition of the trace of function.

The trace of the function
*x*_{1} = *t*_{0}, *t*_{0} ∈ ℝ, is the value of function *ũ* ∈ *C*( ℝ, *ℋ*_{ℰ1}) at the point *t*_{0}.

*If**then for any t*_{0} ∈ ℝ *the estimate**holds.*

*If*

*and*

*then for any t*

_{1},

*t*

_{2}∈ ℝ

*the equality holds*

## 4 The analog of Gauss theorem for rectangle

Let *ℰ* be the ONB of positive trace-class covariation operator **D** of Gaussian measure *γ*. Let Π* _{a,b}* ∈

*𝒦*(

*ℰ*) be a measurable rectangle. For any

*j*∈ ℕ; the equality

*E*= ℝ ⊕

*E*where ℝ = span(

_{j}*e*) and

_{j}*E*= (span(

_{j}*e*))

_{j}^{⊥}. Let

*ℰ*= {

_{j}*e*

_{1},...,

*e*

_{j}_{−1},

*e*

_{j}_{+1},...}. Therefore the following equality

*E*.

_{j}Let
*j* ∈ ℕ; and any *a* ∈ *R* there is the trace *u*|_{x}_{j=}* _{a}* ∈

*ℋ*of the function

_{ℰj}*u*according to the theorem 3.4. Since for any

*j*∈ ℕ; the function

*∂*has the derivative

_{j}u*e*, then according to the theorem 3.4 there is the trace (

_{j}*∂*)|

_{j}u

_{x}_{j}_{=}

*∈*

_{a}*ℋ*.

_{ℰ}_{j}*Let*

**D**

*be a positive trace class operator in Hilbert space E. Let*

*let*Π

*∈*

_{a,b}*𝒦*(

*ℰ*)

*be a measurable rectangle. Then the equality*

*holds for any function*

*Here*

*ϕ*|

_{xj}=

*a*∈

*ℋ*

_{ℰ}_{|}holds for any

*j*∈ ℕ; and any

*a*∈

*R*according to the theorem 15, moreover

*j*∈ ℕ;. Hence the estimates

*j*∈ ℕ;. Let us note that

*x*is unitary operator in the space

_{j}*ℋ*and according to the inequality

_{ℰ}*k*

^{2}≤ 1 +

*k*

^{4},

*k*∈ ℝ, we obtain

Therefore the series in the right hand side (9) converges.

*j*∈ ℕ;. As it shown above, under the summation of the equalities (10) by

*j*∈ ℕ; the series of left hand side absolutely converges. The series of first components in right hand side absolutely converges by definition of the space

## 5 Dirichlet problem for Poisson equation

Let us consider the unique rectangle Π_{0,1} ∈ *𝒦 _{ℰ}*. Let symbol

*L*

_{2}(Π

_{0,1}) notes the subspace of the space

*ℋ*with the support in the set Π

_{ℰ}_{0,1}. Let us note that

*f*=

*χ*

_{Π0,1}

*f*for any

*f*∈

*L*

_{2}(Π

_{0,1}).

_{0,1}. It should be note that for any

*f*∈

*L*

_{2}(Π

_{0,1}) and a given number

*a*≥ 0 we should find a function

To investigate the above problem we apply the variation approach (see [7]). At first we study the space

Let us introduce the trapezoid-like function *ψ _{δ}* : ℝ → ℝ which is given by the equalites

*ψ*(

_{δ}*x*) = 1,

*x*∈ [

*δ*, 1 −

*δ*];

*ψ*‖

_{δ}

_{L}_{2([0,1])}∈ [1 − 2

*δ*, 1],

Let **D** a the nonnegative trace-class operator such that
*e _{j}*} be the ONB of eigenvectors of the operator

**D**and {

*d*} be the corresponding sequence of eigenvalues. For any sequence

_{j}*If**and**then* Ψ_{{}_{δ}_{}} ∈ *ℋ _{ℰ} and the estimates*

*hold.*

_{{}

_{δ}_{}}∈

*ℋ*for the nonnegative function Ψ

_{ℰ}_{{}

_{δ}_{}}is equivalent to the condition

*n*∈ ℕ;. Then

*g*≤ Ψ

_{n}_{{}

_{δ}_{}}≤

*G*and

_{n}*n*∈ ℕ;. Since

_{{}

_{δ}_{}}∈

*ℋ*.

Let us note that 0 ≤ *χ*_{Π}* _{δ}*,

_{1−}

*≤ Ψ*

_{δ}*δ ≤ χ*

_{Π0,1}. Hence

Note that
*k* ∈ ℕ; then the series
*λ* (Π* _{δ}*,

_{1−}

*) > 0. In the other case*

_{δ}*λ*(Π

*,*

_{δ}_{1−}

*) = 0.*

_{δ}*The condition**is necessary and sufficient to the inclusion*

*Ψ*

_{j}_{{}

_{δ}_{}}∈

*ℋ*for any

_{ℰ}*j*∈ ℕ; and the series

*x*∈

*E*, for any

*j*∈ ℕ; where

*j*∈ ℕ; the following equality holds

*∂*Ψ

_{j}_{{}

_{δ}_{}}∈

*ℋ*for any

_{ℰ}*j*∈ ℕ; and

Therefore the series

**Note 21.***Let***D***be a positive operator in the space E such that**is trace class operator. Then there is the sequence**such that**and the condition**satisfies. For example,**k*∈ ℕ.

*Let**Let the sequence**satisfies the condition**Then*

_{{}

_{δ}_{}}

*f*‖

_{ℋ}_{ℰ}≤ ‖

*f*‖

_{ℋ}_{ℰ}according to Cauchy inequality and lemma 1. Since ∂

*(Ψ*

_{j}_{{}

_{δ}_{}}

*f*) = ∂

*Ψ*

_{j}_{{}

_{δ}_{}})

*f*+ Ψ

_{{}

_{δ}_{}}∂

*then*

_{j}fThus we obtain that
_{{}_{δ}_{}}*f*)|_{∂}_{Π0,1} = Ψ_{{}_{δ}_{}}|_{∂}_{Π0,1}*f*|_{∂}_{Π0,1} then (Ψ_{{}_{δ}_{}}*f*)|_{∂}_{Π0,1} = 0 and

*Let**and**Then*

Since {δ} ∈ *l*_{1} then
*f* ∈ *ℋ* then for any *σ* > 0 there is the simple function *g* ∈ *S*_{2}(*E*, *ℛ _{ℰ}*,

*λ*,

_{ℰ}**C**) such that ‖

*f*−

*g*‖

*<*

_{ℋ}*σ*.

*g*has the finite number of values then

Hence for any *σ* > 0 there is a number *ε*_{0} > 0 such that
*ε* ∈ (0, *ε*_{0}). Therefore

The consequence of the lemma 4 is the following statement.

*Let*

**D**

*be a nonnegative operator in the space E such that*

*is trace class operator. Then the set of functions*

*is dense in the space L*

_{2}(Π

_{0,1}).

*a*≥ 0 and

*f*∈

*L*

_{2}(Π

_{0,1}). Let the functional

*Let a* ≥ 0 *and f* ∈ *L*_{2}(Π_{0,1}). *Let**be a stationary point of the functional J _{a,f}. Then u is the solution of Dirichlet problem (11), (12).*

*J*(

_{a,f}*u*+

*tϕ*),

*t*∈ ℝ satisfies the equality

*ϕ*∈

*S*

_{1}. Therefore

*ϕ*∈

*S*

_{1}. Hence

*ϕ*∈

*S*

_{1}according to the theorem 24. Since the set

*S*

_{1}is dense in the space

*ℋ*then the function

*u*satisfies Poisson equation (11). Since

*Let**be the solution of Dirichlet problem (11), (12). Then it is the critical point of the functional (13).*

*J*(

_{a,f}*u*+

*t*),

_{ϕ}*t*∈ ℝ, has the derivative

Since
*u* is the stationary point of the functional *J _{a,f}*.

The inequality
*f* ∈ *H*. Then for any
*c* ≤ ‖ *f* ‖* _{ℋ}*. Then according to R theorem there is the element

*a*> 0. The space

*Let a* > 0 *and f* ∈ *L*_{2}(Π_{0,1}). *Then the functional (13) has the unique point of the minimum in the space*

Let *f* ∈ *ℋ _{ℰ}*. Then for any

*c*≤ ‖

*f*‖

*. Then according to Riesz theorem for any*

_{ℋℰ}*a*> 0 there is the element

Therefore the functional *J _{a,f}* has the unique point of the minimum in the space

*Let a* > 0 *and f* ∈ *L*_{2}(Π_{0,1}). *Then the function**is point of minimum of the functional (13) if and only if it is the generalized solution of Dirichlet problem (11), (12).*

If the function
*u* is is the generalized solution of Dirichlet problem (11), (12).

*u*is is the generalized solution of Dirichlet problem (11), (12). Then the right hand side of the expression (14) is equal to zero. Therefore for any

Hence the function

## 6 Conclusions

In this paper we show that the theory of Sobolev spaces and its application to partial differential equation can be constructed for the function on domains in infinite dimension Hilbert space endowing with finite additive shift invariant measures. We study the class of finite additive shift invariant measures on the real separable Hilbert space *E*. For any choice of such a measure we consider the Hilbert space *ℋ* of complex-valued functions which are square-integrable with respect to this measure. Some analogs of Sobolev spaces of functions on the space *E* are introduced. The analogue of Gauss theorem is obtained for the simplest domains such as the rectangle in the space *E*. The correctness of the problem for Poisson equation in the rectangle with homogeneous Dirichlet condition is obtained and the variational approach of the solving of this problem is constructed.

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