The Cauchy Problem for the Heat Equation with a Random Right Part from the Space ( ) Sub φ Ω

The influence of random factors should often be taken into account in solving problems of mathematical physics. The heat equation with random factors is a classical problem of the parabolic type of mathematical physics. In this paper, the heat equation with random right side is examined. In particular, we give conditions of existence with probability, one classical solutions in the case when the right side is a random field, sample continuous with probability one from the space ( ) Subφ Ω . Estimation for the distribution of the supremum of solutions of such equations is founded.


Introduction
The subject of this work is at the intersection of two branches of mathematics: mathematical physics and stochastic processes.
The physical formulation of problems of mathematical physics with random factors was studied by Kampe de Feriet [1].In the works [2] and [3], a new approach studying the solutions of partial differential equations with random initial conditions was proposed.The authors investigate the convergence in probability of the sequence of function spaces of partial sums approximating the solution of a problem.The mentioned approach was used in the works [4]- [7].In the paper [3], the application of the Fourier method for the homogeneous hyperbolic equation with Gaussian initial conditions is justified.The conditions of the existence of the classical solution of this equation in terms of correlation functions are also studied.Homogeneous hyperbolic equation with random initial conditions from the space ( ) Sub ϕ Ω is considered in [8]- [11].The model of a solution of a hyperbolic type equation with random initial conditions was investigated in the papers [12] [13].
There is a study on a boundary-value problem of mathematical physics for the inhomogeneous hyperbolic equation with ϕ -subgaussian in right part [8] [14].The parabolic type equations of Mathematical Physics with random factors of Orlicz spaces have been studied in the papers [15] [16].Further references can be found in [8] [17]- [21].
We consider a Cauchy problem for the heat equations with a random right part.We study the inhomogeneous heat equation on a line with a random right part.We consider the right part as a random function of the space ( ) Sub ϕ Ω .The Gaussian stochastic process with zero mean belongs to ( ) [22].The conditions of existence with probability one of the classical solution of this problem are investigated.For such a problem has been got the estimation for the distribution of the supremum solution.
The paper consists of the introduction and three parts.Section 2 contains necessary definitions and results of the theory of the ( ) Sub ϕ Ω space.In Section 3, we consider heat equations with random right-hand side.For such problem conditions of existence, with probability one, of classical solution with random right-hand side from the space ( ) p L Ω are found.The estimation for distribution of supremum of this problem has been got in Section 4.
Remark 1. [24] The Gaussian stochastic process ( ) X t with zero mean belongs to ( ) A Family of Strongly ( ) Random Variables and a Family Strongly ( ) Definition 6. [21] The random variable ( ) Properties and applications of ( ) SSub ϕ Ω random variables and stochastic processes from ( ) SSub ϕ Ω can be found in [21].Definition 7. [7] A family ∆ of random variables ξ of the space ( ) where I is at most countable and i i ξ ∈ ∆ , i I ∈ .Theorem 1. [7] Let ∆ be a strongly ( ) Sub ϕ Ω family of random variables.Then the linear closure ∆ of the family ∆ in the space ( )

2
L Ω and in the mean square sense is a strongly ( ) Sub ϕ Ω family.Definition 8. [21] The stochastic process be a family of jointly strongly ( ) =∞ is a family of measurable functions in ( ) , , T θ µ and the integral ( ) ( ) ( ) is well defined in the mean square sense, than the family of random variables and assume that the partial derivatives , Let there exist a monotone increasing continuous function ( ) for all z and for sufficiently small 0 ε > where ( ) ( ) ( ) with probability one the partial derivatives , , i j m =  , exist and are continuous.

The Heat Equations with Random Right Part
We consider the Cauchy problem for the heat equation Let the function ( ) is a random field sample continuity with probability one from the space ( ) , , , B x t z s be a continuous function.Problem when the function ( ) nonrandom has been seen in [25].
ξ is a random field, sample continuity for each 0 t > with probability one, there is a continuous derivative ( ) Then for the function ( ) , we deduce that the integral ( ) ∫ exist with probability one, and therefore the integral , and therefore it implies from [26] that the integral Fourier transform exist, and the inverse integral Fourier transform Theorem 5. Let the conditions of Lemma 4 be satisfied and By deriving (7) with respect to x and t , we easily see that ,  , 4  , , ,  d d  ,  ,  , , ( ) Then Lebesgue integrals ( )

∫
For existence of this integral with probability one it is enough to prove that there exists following integral ( ) Integrating by parts and using the conditions of the lemma, we obtain for y ≠ 0
Proof.Indeed, it is easy to show that the function for some − ≤ , then for sufficiently small h inequa- lity ( 9) and (10) will have the form where ( ) ( ) ( )  (11) holds for all 0 ε > ( ) ( ) Condition ( 12) holds if ( ) ) According to Theorem 5 to make the function ( ) , u x t be the solution of problems ( 1) and ( 2) it is sufficient to prove that integrals ( 5) and ( 6 Using generalized Minkovskoho inequality we obtain and for sufficiently small h , using the inequality (10), we have ( ) and for sufficiently small h , using the inequality (9), we have Thus we obtain from ( 14), ( 15), ( 16) and ( ( ) are some constants.Consider

Corollary 1 .
Let in the conditions of Lemma 6 the function ( ) ( ) δ > we obtain the inequality 8.
field, sample continuous with probability one from the ( ) Sub ϕ Ω and the conditions of Lemma 4 and Lemma 5 hold, Then the function ( ) , u x t which is represented in the form (4) is a classical solution to the problems (1) and(2).Proof.This theorem follows from Theorems 5 and 3.

For
field, sample continuous with probability one from the space ( ) SSub ϕ Ω , where ( ) ) converge uniformly in probability in x A ≤ , 0 t T ≤ ≤ to the integrals According to Theorem 6, using the Example (1), to make integral (5) and (6) converge in probability in( )C T  the following conditions must hold

(
This theorem follows from Theorem 8. Then the function ( ) )