Solution to a Parabolic Differential Equation in Hilbert Space Via Feynman Formula - I

A parabolic differential equation $u_t(t,x)=Lu(t,x)$ is considered, where $t\geq 0$ is time, $x$ is a coordinate that belongs to a finite- or infinite-dimensional real separable Hilbert space, and $L$ is a second-order differential (with respect to $x$) operator with time-independent coefficients that may depend on $x$. Assuming the existence of a strongly continuos resolving semigroup for this equation, we construct a representation of this semigroup by means of a Feynman formula, i.e. we present it in the form of the limit of a multiple integral over Hilbert space as the multiplicity of the integral tends to infinity. This representation gives a unique solution of the Cauchy problem in the uniform closure of the set of smooth cylindrical functions on the Hilbert space. Moreover, this solution depends continuously on the initial condition. In the case where the coefficient in front of the first derivative term in $L$ vanishes we prove that there exists the strongly continuous resolving semigroup (this implies the existence of the unique solution to the Cauchy problem in the class mentioned above) and that the solution to the Cauchy problem depends continuously on the coefficients of the equation.


Introduction
Representation of a function by the limit of a multiple integral as multiplicity tends to infinity is called a Feynman formula, after R.P. Feynman, who was the first to use such representations on the physical level of rigor for the solution of the Cauchy problem for PDEs [24,25]. The term "Feynman formula" in this sense was introduced in 2002 by 338 Моделирование и анализ информационных систем Т. 22, № 3 (2015) O.G. Smolyanov [31]. One can find out more about the research into Feynman formulas up to 2009 in [33]. The most recent (but not complete) overview is [37] (2014, in Russian). It is important to note that Feynman formulas are closely related to Feynman-Kac formulas [30], however the latter will not be studied in the present article. Usage of Feynman and Feynman-Kac formulas includes exact or numerical evaluation of integrals over Gaussian measures on spaces of high or infinite dimension; some useful approaches to this topic are developed in [6,8].
Differential equations for functions of an infinite-dimensional argument arise in (quantum) field theory and string theory, theory of stochastic processes and financial mathematics. Evolutionary equations (i.e. PDEs in the form ′ ( , ) = . . . ) in infinite-dimensional spaces have been studied since 1960s by O.G. Smolyanov, E.T. Shavgulidze, E. Nelson, A.Yu. Khrennikov, S. Albeverio and others. We will mention just some of the publications, which are most recent and relevant for our study.
In [3] the Schrödinger equation in Hilbert space is studied. The equation includes the terms of second, first and zero order, the coefficient of the second order term is constant. The solution to the Cauchy problem is given by a Feynman-Kac-Ito formula.
In [22] a solution to a heat equation in Hilbert space without the terms of the first and zero order is discussed, the coefficient of the second-derivative term is constant. The solution is given in the form of a convolution with the Gaussian measure (analogous to the finite dimensional equation with constant coefficients), the existence of the resolving semigroup is proved. In [14] the solution to the same equation is given by a Feynman-Kac formula.
In [15] the parabolic equation in finite-dimensional space is studied for the case of variable coefficients. Under the assumption that a strongly continuous resolving semigroup exists for the Cauchy problem, Feynman and Feynman-Kac formulas were proven in [15] for the solution.
In [28], for a class of equations in an infinite-dimensional space, with a variable coefficient at the highest derivative (but without first-and zero-order derivatives' terms), a Feynman formula was obtained and the existence of resolving semigroup was proven.
In spaces over the field of p-adic numbers, Feynman and Feynman-Kac formulas for the solutions of the Cauchy problem for evolutionary equations were given in [11,12].
In [19,20], Schrödinger and heat equations in R were studied in the case of timedependent coefficients, and a Chernoff-type theorem was proven for this case.
The present article extends my first results in this area [28] to the case of non-zero coefficients at the first-and zero-order derivatives.
I do not provide the technical proofs for the two key theorems to keep the paper short, but give all the background that is relevant for the proofs. The article may be used as a very short introduction to analysis in Hilbert space and to the applications of 0 -semigroup theory in solving evolutionary PDEs.

Notation and definitions
The symbol stands for the real separable Hilbert space with the scalar product ⟨·, ·⟩. The self-adjoint, positive, non-degenerate (hence injective), linear operator : → Solution to a parabolic differential equation in Hilbert space . . .

339
is assumed to be defined everywhere on . The operator is assumed to be of trace class, which means that for every orthonormal basis ( ) in the sum ∑︀ ∞ =1 ⟨ , ⟩ = tr is finite; this sum is called the trace of (it is independent of the choice of the basis ( )).
The symbol below stands for any complex Banach space. The symbol ( , ) stands for space of all linear bounded operators in , endowed with the classical operator norm.
Symbol ( , ) will mean the set of all continuous functions from to , where and are topological spaces. A function : → R is called cylindrical [5,10], if there exist vectors 1 , . . . , from and function : R → R such that for every ∈ the equality ( ) = (⟨ , 1 ⟩ , . . . , ⟨ , ⟩) holds. In other words, the function : → R is cylindrical if there exists an −dimensional subspace ⊂ and orthogonal projector : → such that ( ) = ( ) for every ∈ . The cylindrical function can be imagined as a function, which is first defined on and then continued to the entire space in such a way that ( ) = ( 0 ) if 0 ∈ and ∈ ( 0 + ker ). Symbol = ∞ , ( , R) stands for the space of all continuous bounded cylindrical functions → R such that they have Fréchet derivatives [17] of all positive integer orders at every point of , and their Fréchet derivatives of any positive integer order are bounded and continuous.
Symbol ( , R) stands for the Banach space of all bounded continuous functions → R, endowed with a uniform norm ‖ ‖ = sup ∈ | ( )|. It is regarded as a closed subspace of a complex Banach space ( , C). Let = ∞ , ( , R) be the closure of the space in ( , R). It is clear, that with the norm ‖ ‖ = sup ∈ | ( )| is a Banach space, as it is a closed linear subspace of the Banach space ( , R). Function belongs to if and only if there is a sequence of functions ( ) ⊂ such that lim →∞ = , i.e. lim →∞ sup ∈ | ( ) − ( )| = 0. Symbol ( , ) stands for a Banach space of all bounded continuous functions : → , endowed with the uniform norm ‖ ‖ = sup ∈ ‖ ( )‖.
Let be the closure of in ( , ). If ∈ , and : → is linear, trace class, positive, non-degenerate operator, then symbol stands for the Gaussian probabilistic measure [1,5,34] on with expectation and correlation operator , i.e. the unique sigma-additive measure on Borel sigmaalgebra in such that the equality ∫︀ )︀ holds for every ∈ . To make it shorter, we will write instead of 0 . See section 3.1. for useful formulas about integration over the Gaussian measure.
If : → is a vector field, and : → R and : → R are real-valued functions, then symbol defines a differential operator on the space of functions : → R The pair (ℒ, ) defines a linear operator ℒ with the domain . It will be shown in theorem 4.2 that ( ) ⊂ when , , and have certain properties. So ( , ) is a densely defined (on ) operator : ⊃ → . Here the earlier defined spaces and are endowed with the uniform norm, induced from ( , R). Let ( , 1 ) be the closure of ( , ) in . This means that Remark 2.1. Further, in theorem 4.1, we will prove that for every ≥ 0 and for , , and having certain properties the following holds i) ( ) ⊂ , ii) operator is bounded, and iii) ⃒ ⃒ =0 = for all ∈ . This will allow us to use the Chernoff approximation (theorems 3.1, 3.2) and prove the main result of the present article, theorem 4.4. → R is cylindrical and measurable, i.e. ( ) = (⟨ , 1 ⟩, . . . , ⟨ , ⟩) for some ∈ N, some measurable function : R → R, and some finite orthonormal family of vectors 1 , . . . , from space , then

Lemma 3.2. (Explicit form of some integrals over Gaussian measure)
Let be a real separable Hilbert space of finite or infinite dimension,̃︀: → be a linear, trace class, symmetric, positive, non-degenerate operator,̃︀ be the centered Gaussian measure on with the correlation operator̃︀, and : → be a bounded linear operator. Let and be non-zero vectors from .
Then the following equalities hold: Proof. Formulas (3) and (4) can be found in [5], chapter II, §2, 1 ∘ . Formula (5) can be derived from the fact that the function under the integral is cylindrical, so lemma 3.1 can be employed. For a proof of (6), one can make the change of variable in the , and the integral reduces to (3). → is positive, non-degenerate, trace class, and self-adjoint. We will identify with the symbol the centered Gaussian measure on with the correlational operator . Let > 0; the symbol denotes operator, that takes ∈ to ∈ . Let : → R be a continuous integrable function. Then Proof uses the uniqueness of the Gaussian measure with a given Fourier transform, and the standard theorem of changing variable in the Lebesgue integral.
Lemma 3.4. (On integrability of a polynomial multiplied by an exponent) Let , , be as above, : R → R be a polynomial, and ∈ R.
Proof is easy to construct by relying on Fernique's theorem [23], which (applied to this case) says that there exists such > 0 that ∫︀

Derivatives of cylindrical functions
Proposition 3.1. Let be a cylindrical real-valued function on , i.e. there is a number ∈ N and a function : R → R such that for every ∈ the equality ( ) = (⟨ , 1 ⟩, . . . , ⟨ , ⟩) holds. A set of vectors 1 , . . . , can be considered orthonormal without loss of generality. Lets complete this set to an orthonormal basis ( ) ∈N in .
Моделирование и анализ информационных систем Т. 22, № 3 (2015) Then: 1. Function is differentiable in the direction ℎ if and only if the function is differentiable in the direction (⟨ℎ, 1 ⟩, . . . , ⟨ℎ, ⟩) ∈ R , and where the symbol defines the partial derivative with respect to the -th argument of the function , and ( 1 , . . . , , 0, 0, 0, . . . ) = 1 1 + · · · + . If the function has a Fréchet derivative at the point , then ′ ( ) is a vector whose first coordinates yield the gradient of the function , and the other coordinates are zero: 2. Function has a Fréchet derivative in if and only if the function has a Fréchet derivative in R .
3. Let : → be a trace-class operator (i.e. let tr < ∞). Then where is the matrix of the operator in the basis 1 , . . . , , where is the projector to the linear span of the vectors 1 , . . . , .
Proof is a straight-forward application the derivative's definition.

Differential operator on a finite-dimensional space
is the class of all bounded real-valued functions on R , which have bounded partial derivatives of all orders. Suppose also that ( ) ≤ 0 for all ∈ R .
For ∈ ∞ (R , R) we define a differential operator by the formula Suppose that there exists a constant κ > 0 such that for every = ( 1 , . . . , ) ∈ R and all ∈ R the ellipticity condition is fulfilled: Take an arbitrary constant > 0 and function ∈ ∞ (R , R). Then: 1. There is a unique function ∈ ∞ (R , R), which is a solution of the equation 2. For every function ∈ ∞ (R , R) the following estimate is true Solution to a parabolic differential equation in Hilbert space . . .
Note that equation (10) can have unbounded solutions; this does not contradict the lemma.

Strongly continuous semigroups of operators and evolutionary equations
Let be a complex Banach space.
Definition 3.1. By a strongly continuous one-parameter semigroup ( ) ≥0 of linear bounded operators in we (following [26,18]) mean the mapping of the non-negative half-line into the space of all bounded linear operators on , which satisfies the following conditions: Definition 3.2. By the generator of a strongly continuous one-parameter semigroup ( ) ≥0 of linear bounded operators on we mean a linear operator ℒ : ⊃ (ℒ) → given by the formula where the limit is understood in the strong sense, i.e. it is defined in terms of the norm in the space .
The use of the symbol ℒ for the generator is related to the fact that the generator is always a closed operator: [26], p. 51) The generator of a strongly continuous semigroup is a closed linear operator with a dense domain. The generator defines its semigroup uniquely.
2. For every 0 ∈ there is a unique mild solution to abstract Cauchy problem (12), which is given by the formula ( ) = ( ) 0 . Proposition 3.5. (On the closability of a densely defined dissipative operator ) (proposition 3.14 in [26]) A linear dissipative operator ℒ : The main tool for the construction of Feynman formulas for the solutions of the Cauchy problem is Chernoff's theorem. For convenience we decompose its conditions into several blocks and give them separate names, as follows. (CT3). There exists a dense subspace ⊂ such that for every ∈ there exists a limit ′ (0) = lim →0 ( ( ) − )/ = ℒ .
Remark 3.1. There are several slightly different definitions of the Chernoff equivalence, see e.g. [36,29,37]. We will just use this one not going into details. The only thing we need from this definition is that if satisfies all the conditions of Chernoff's theorem, then by Chernoff's theorem the mapping is Chernoff-equivalent to the mapping 1 ( ) = ℒ , i.e. the limit of ( ( / )) as tends to infinity yields the 0 -semigroup ( ℒ ) ≥0 .    is invariant with respect to them, the above theorems about are applicable to .

Properties of spaces , , 1
Remark 3.4. It directly follows from the definitions of these spaces that i) ⊂ 1 ⊂ ⊂ ( , R) ⊂ ( , C); ii) and 1 are dense in ; iii) is a Banach space.
Proof. It follows from the definition of the space that the function ∋ : → R is bounded and its Fréchet derivatives of all orders exist and are bounded. In particular, there exists sup ∈ ‖ ′ ( )‖ = < ∞. For every ∈ and every ∈ one can see the estimate which implies the uniform continuity of .
Solution to a parabolic differential equation in Hilbert space . . . It is shown in [28] that if is infinite-dimensional, then a non-constant function that belongs to cannot have a limit at infinity. For example, the function ↦ −→ exp(−‖ ‖ 2 ) belongs to ( , R) but not to .
Remark 3.6. Suppose that : R → R is a family of infinitely-smooth functions, uniformly bounded with their first and second derivatives: For example, ( ) = sin( ( − )), where and are constants and 0 < ≤ 1.
Then function belongs to the class 1 . This statement can be easily extended to the case : R → R.
Remark 3.7. Space is not separable (it does not have a countable dense subset). In the case of one-dimensional it can be shown similar to the standard proof of the nonseparability of (R, R). If dim > 1, then R 1 can be embedded into as a linear span of a non-zero vector ∈ . Using this, one can embed the set of cylindrical functions contributing to the non-separability of in the case of one-dimensional , into the space in the general case.
Remark 3.8. By Remark 3.7 and the inclusion ⊂ 1 ⊂ , one can see that 1 and are not separable too.

Family provides a semigroup with generator
Then: 1. If ≥ 0 and ∈ ( , R), then ∈ ( , R). For every ≥ 0 the operator : ( , R) → ( , R) is linear and bounded; its norm does not exceed ∈ , ∈ , then the space for every ≥ 0 is invariant with respect to the operator .
3. If ∈ , ∈ , ∈ , then the space for every ≥ 0 is invariant with respect to the operator .

If
Analogue of theorem 4.1 for finite-dimensional can be found in [15]. The proof for the case of infinite-dimensional follows the same general line but is more involved. It uses lemmas from sections 3.1., 3.5. and will be published in a separate paper. Symbol stands for the identity operator. Then: 1. If ∈ , ∈ , ∈ and ∈ , then ∈ . If ∈ , ∈ , ∈ and ∈ , then ∈ . 2. If ∈ , ∈ , ∈ , then for each > 0 the operator − is surjective on , therefore ( − )( ) = is a dense subspace in .
The proof of theorem 4.2 is based on the results of sections 3.2., 3.3. and will be published in a separate paper. Item 1 follows from the definition of the operator . Items 2 and 3 are derived from lemma 3.5, proposition 3.1 and proposition 3.5. Item 4 is derived from item 2. Item 5 is obtained by proceeding to the limit in the dissipativity estimate proven in item 3 and then applying proposition 3.5. Suppose that ∈ , ∈ , ∈ , and for every ∈ we have ( ) ≥ 0 ≡ const > 0 and ( ) ≤ 0. As ∈ , there exists a sequence ( ) ⊂ , converging to uniformly; let us additionally claim that this sequence can be selected in such a way that ( ) ≤ 0 for all ∈ N and all ∈ . Then the following holds: 1. If the closure ( , 1 ) of the operator ( , ) is a generator of a strongly continuous semigroup (︁ )︁

≥0
of linear continuous operators on the space , then where limit exists for every ∈ and is uniform with respect to ∈ [0, 0 ] for every uniformly with respect to ∈ and ∈ [0, 0 ]. Proof.

Feynman formula solves the Cauchy problem for the parabolic equation
We want to find a function : [0, +∞) × → R satisfying the following conditions (we call them Cauchy problem for the parabolic differential equation): To this Cauchy problem, we relate the so-called abstract Cauchy problem (see Definition 3.3), which we define as the following system of conditions upon the function : [0, +∞) → : Using this correspondence, we start from Definition 3.3 and define the solution of problem (20).
We call a function : [0, +∞) × → R a strong solution of problem (20) if it satisfies the following conditions: Definition 4.2. We call a function : [0, +∞) × → R a mild solution of problem (20) if it satisfies the following conditions: Finally, let us state and prove the main result of the article. We use definitions and notation from Section 2. Suppose ∈ , ∈ , ∈ . Suppose there is a number 0 > 0 such that for all ∈ we have ( ) ≥ 0 and ( ) ≤ 0. As ∈ , there exists a sequence ( ) ⊂ , converging to uniformly; let us additionally require that this sequence can be selected in such a way way that ( ) ≤ 0 for all ∈ N and all ∈ .
Then the following holds: 1. If there exists a strongly continuous semigroup with the generator , then for every 0 ∈ 1 there exists a solution of problem (22), unique in the class ([0, +∞), ).
The solution depends continuously on 0 , and is given by the formula where the limit is uniform with respect to ∈ [0, 0 ] for every 0 > 0.
4. Let = 0, and let the functions ∈ , ∈ and ∈ be given for all ∈ N. Let = 0 for all ∈ N. Suppose there exists 0 > 0 such that ( ) ≥ 0 and ( ) ≤ 0 for all ∈ N and all ∈ . Let us use the symbol for the operator that corresponds to the functions , and , and the symbol 0 for the operator that corresponds to the functions , and . Suppose also that ( ) → ( ) and ( ) → ( ), uniformly with respect to ∈ . We denote as the solution of problems (22) and (23) for the operator . For solution of problems (22) and (23) with the operator , we use the symbol .
Then ( , ) converges to ( , ) as → ∞, uniformly with respect ∈ and uniformly with respect to ∈ [0, 0 ] for every fixed 0 > 0.  Analogous theorems for C-or R -valued functions can be formulated mutatis mutandis. The result will hold true due to the theorem above and the linearity of and . The only additional condition will be that the coefficients of the equation must be real-valued. The same remark is applicable to all the key theorems of this article.
Proof of the theorem. 1. Suppose that there exists a strongly continuous semigroup with the generator . Then by item 1 of proposition 3.4 we obtain the existence of a strong solution (definition 3.3) to Cauchy problem (21), and the solution is unique in the class ([0, +∞), ). By item 1 of theorem 4.3 the semigroup is given in the form described. Using the relation between problems (20) and (21) explained in remark 4.1, we obtain the solution for problem (22). The solution is unique in the class ([0, +∞), ), as follows from remark 4.1.
2. The proof is similar to that in item 1. The only difference is that in proposition 3.4 we use item 2 instead of item 1.
3. The existence of the sought semigroup follows from item 2 of theorem 4.3. The estimate for the supremum of the absolute value of the solution follows from the fact that the semigroup is contractive.
Let us explain how the equality ( , ) = lim →∞ (︁(︁ )︁ 0 )︁ ( ) implies formula (24). For a continuous bounded function : → R and a point ∈ , the following change of variables rule in the integral is correct: Applying this rule, and changing to 2 ( ) , we come to the equality In the same way expressions for > 2 are derived. Thus, the formula (24) is proven. 4. The proof follows immediately from item 3 of theorem 4.3.