Feynman Formulas Representation of Semigroups Generated by Parabolic Difference-Differential Equations

We establish that the Laplas operator with perturbation by symmetrised linear hall of displacement argument operators is the generator of unitary group in the Hilbert space of square integrable functions. The representation of semigroup of Cauchy problem solutions for considered functional differential equation is given by the Feynman formulas.


Introduction
In this paper we investigate the questions of correct resolvability of Cauchy problem for modeling parabolic differential-difference the equations of the form supplemented with the initial data Here , for any , u 0 is a given function and . The equations of this form arise at the description of the phenomena of diffusion or heat conductivity with the sources, nonlocally dependent on the state .Physically the state u means the density distribution of the concentration or the temperature.In particular, the equations of a kind (1) arise at research of problems of management by the phenomena of a heat transfer in which dynamics of a state is given by the differential equation u with management g.We obtain the Equation (1) in case when management g is given by the action on a state function u of a deviation argument operators in a composition with operators of differentiation and multiplication by the function (see [1][2][3]).
In this work we obtain the representation of semigroup solutions of the Cauchy problem for the functionaldifferential equation through the Feynman formula (see [4]).It means that although the representation of the evolutionary operator of the Cauchy problem (1) can be defined only in terms of the spectral decomposition (in the simplest situation in terms of the Fourier transform of the solution), nevertheless we obtain an approximation of the evolutionary operator by sequence of n-fold compositions of integrated operators which kernels are elementary functions.
In the terms of the monography [1], differentialdifference Equation (1) concerns to type mixed differential-difference equation without a deviation on time.Nonlinear parabolic differential-difference the equations arise in the investigation of nonlinear optical systems with a feedback (see [3]).In work [5] the mixed problem for nonlinear parabolic differential-difference equations had been formulated.Also it was established what properties distinguish the specified problem from the mixed problem for parabolic differential equations.The linear Cauchy problem (1), (2) can be considered as a linearization of specified nonlinear mixed problems.
Firstly we prove the correctness of Cauchy problem (1), (2) by using of Fourier transformation and obtain the representation of Fourier image of its solution.After that we construct the approximation of solution by Feynman formulas.We extend the obtained result onto the operators with distributed deviation of the space argument.
The obtained results gives not only the expression of the decision of Cauchy problem with the help of constructive algorithms, but also the investigation of probabilistic structure of the phenomena reject the arguments in the heat equation.The above Feynman formulas define the approximation of a Markov random process, such that the mean value of some functional on this process is the solutions of the Cauchy problem.

Correct Resolvability of the Cauchy Problem and Generation of Semigroup Operators
Let us determine the solution of the Equation (1), satisfying the initial condition (2).Definition.A strong solution of the Cauchy problem (1), (2) call the function which Satisfies the equation (1) and condition (2).Function 3) the sequence of functions converges to fun- Note that both strong and weak solution satisfies the Equation (1) and condition (2) in the sense of the integral identity.
Suppose that the existence of solution of Cauchy problem (1), ( 2) is obtained.To find a representation of the solution of the Cauchy problem (1), (2) through the initial condition, we apply the Fourier transform F to the left and right part of Equation (1): Let the function be the Fourier transform in the first variable.Then Equation (3) takes the form: , and the initial condition (2) transforms into the equation: Then by using of the equations and the initial condition we obtain that Thus, established the following statements Then the strong continuity at any point follows from the semigroup property.Moreovet the type of above semigroupe  is equal to the value  , , 0 t t  0 In fact, according to the proposition 1, if the solution of the Cauchy problem (1), (2) exists, then it representable in the form Conversely, if function is defined by the equality (4) then the function , satisfy the equality ( 5), the equality (1) and the condition (2).In fact, if the function Then, by the unitarity of the Fourier transform, for each initial function there is the function (5) which is a strong solution of the Cauchy problem (1), (2).Hence, the formula (5) defines the image of the S is the shift operator on value .h R  The obtained representation of the solution of the Cauchy problem (1), ( 2) is not constructive.For approximation of the solution obtained by using of sequences of multiple integrals of elementary functions we use the approach from papers [4,6,7] based on the Feynman formulas. (6)

Chernoff's Approximation of Solutions of the Cauchy Problem
which is satisfied for arbitrary function from the dense in space . The proposed form of Chernoff's approximation of the semigroup related to the fact that the first term in formula ( 6) corresponds to the dynamics, generated by the unperturbed heat equation, and the second and third terms for small values of the variable presente the influence of the displaced sources.
t Following the approach offered in [4], we define the operator-valued function equivalent in Chernoff sense to the semigroup of operators Chernoff's theorem (see [8]) states that: . Then for any u H  and any the By following the definition in [4] the operator-valued function will be called equivalent by Chernoff semigroup if for any  , 0, To prove of lemma 1 we compute the function

y h x y h x y h x y h u y y
For arbitrary and for given in Equation ( 1) we define the operator function assuming that for each function its image defined by equality Hence, by lemma 1 on linear manifold D the operator coincides with the generator of the semigroup  , 0.
, is continuous in the strong operator topology on the semiaxis and supposes an esti-0 t  mation on norm Firstly we prove this statement for the case , where The operators h  S are bounded and have unit normed.Hence, the operator-valued function is continuous in the strong operator topology and satisfy the estimate Therefore the operator-valued function where K is some even function of space and the remaining terms are defined in the consideration of the Equation (1).Hence Fourier transform  by applying Fourier transform F to the equality (8).Hence we obtain, that and therefore Therefore the following analogue of Theorems

Theorem 2 .
The Cauchy problem (1), (2) has a unique generalized solution  , 0, u t t    U which is defined as the action of the semigroup on the initial condition .

4. Some Generalization on the Case of Distributed Deviation of the Space Argument
So that, to find the approximate solution of the Cauchy problem by Feynman formula, we define, generalizing the formula (6), operator-valued function 0