The Cauchy Problem for the Heat Equation with a Random Right Part from the Space Subφ (Ω) ()
1. 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 
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 
-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
. 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 
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 
are found. The estimation for distribution of supremum of this problem has been got in Section 4.
2. Random Processes from 
Space
Definition 1. [23] An even continuous convex function
, 
such that 
and 
for 
and 
is called an 
function.
Definition 2. [21] We say an 
function 
satisfies the 
condition if there exist constants
, 
, 
such that 
for all
,
.
Lemma 1. [21] Let 
be an 
function. Then 1. 
for 
and
;
2. 
for 
and
;
3. 
for
;
4. The function 
is non decreasing for
.
Lemma 2. [21] Let 
be the inverse to an 
function 
for
. Then 
is a convex increasing function such that 1. 
for 
and
;
2. 
for 
and
;
3. 
for
;
4. the function 
is nonincreasing for
.
Definition 3. [23] Let 
be an 
function. The function 
is called the YoungFenchel transform of the function
. The function 
is an 
function as well.
Let 
be a standard probability space.
Definition 4. [21] Let 
be an 
function for which there exist constants 
and 
such that 
for
. The set of random variables
, 
, is called the space 
generated by the 
function 
if 
and there exists a constant 
such that 
for all
.
The space 
is a Banach space with respect to the norm [21] 
.
Definition 5. [23] The stochastic process 
belongs to space
, 
if 
for all
.
Remark 1. [24] The Gaussian stochastic process 
with zero mean belongs to
, where 
and
.
A Family of Strongly 
Random Variables and a Family Strongly 
Stochastic Processes
Lemma 3. [21] If
, then there exists a constant 
such that
.
Definition 6. [21] The random variable 
is called strongly
, 
random variable if
.
Properties and applications of 
random variables and stochastic processes from 
can be found in [21] .
Definition 7. [7] A family 
of random variables 
of the space 
is called 
family if
for all
, where 
is at most countable and
,
.
Theorem 1. [7] Let 
be a strongly 
family of random variables. Then the linear closure 
of the family 
in the space 
and in the mean square sense is a strongly 
family.
Definition 8. [21] The stochastic process 
is called an 
process if the family of random variables 
is an
.
Theorem 2. [7] Let 
be a family of jointly strongly 
stochastic processes. Then 
is a measurable space. If 
is a family of measurable functions in 
and the integral 
is well defined in the mean square sense, than the family of random variables 
is an 
family.
Theorem 3. [9] Let 
be the 
-dimensional space,
, 
,
.
. Assume that the process 
is separable and
where 
is a monotone increasing continuous function such that 
as
. We also assume that
,where 
and 
is the inverse function to
. If the processes 
converge in probability to the process 
for all
, then 
converge in probability in the space
.
Theorem 4. [9] Let 
and let
, 
, be a separable random field such that
. Put 
and assume that the partial derivatives
and
exist. Let there exist a monotone increasing continuous function
, 
, such that 
as 
for
, 
, 
and
,
. Assume that
If 
for all 
and for sufficiently small 
where
, then with probability one the partial derivatives
, 
, 
, exist and are continuous.
3. The Heat Equations with Random Right Part
We consider the Cauchy problem for the heat equation
(1)
, subject to the initial condition
(2)
Let the function 
is a random field sample continuity with probability one from the space
, such that
,
. Let us denote
. Let 
be a continuous function. Problem when the function 
nonrandom has been seen in [25] .
Lemma 4. Let 
is a random field, sample continuity for each 
with probability one, there is a continuous derivative 
for 
and satisfy condition
(3)
Then for the function 
for each 
the integral Fourier transform
exist and
.
Proof. Since, by Fubini’s theorem, 
, 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
exist.
Theorem 5. Let the conditions of Lemma 4 be satisfied and
and
(4)
If the following integrals exist
, 
,
and for all 
and 
there exists a sequence
, 
for
, such that the sequence of integrals
(5)
(6)
converges in probability, uniformly for
, 
, then 
is the classical solution to the problem (1) and (2).
Proof. Since the integrals (5) and (6) converges in probability uniformly for
, 
, there exists a subsequence
, 
as
, such that
converges with probability one to
uniformly for
, 
, Let
(7)
By deriving (7) with respect to 
and
, we easily see that
Since for 
converges to
, and 
converges to 
uniformly for
, 
with probability one, we conclude that 
satisfies Equation (1).
Indeed, 
Lemma 5. [9] Let 
be a random field, sample continuity from the space
. Let 
be the correlation function of the field
. For all 
assume that:
1. The derivatives 
exist;
2 
3.
, at 
or
.
Then Lebesgue integrals
exist with probability one.
Proof. We shall prove the existence of the integral
For existence of this integral with probability one it is enough to prove that there exists following integral
There is an inequality
Consider
Integrating by parts and using the conditions of the lemma, we obtain for y ≠ 0
Then
Therefore
for
. The latter integral converges under
. The existence of integrals
, 
can be proved similarly.
Lemma 6. [15] Let a function
, 
and 
be such that:
1. 
2. 
for all
,
. Let
, 
be a continuous increasing function such that 
for all
, and the function 
is increasing for
, and for some constant
.
Then 
for all 
and
.
Corollary 1. Let in the conditions of Lemma 6 the function
, 
,
. Then
(8)
for all
.
Proof. Indeed, it is easy to show that the function
increases with
. Therefore in Lemma 6 taking function 
we obtain the inequality 8.
Corollary 2.
. (9)
. (10)
for some
.
Remark 2. If in the conditions of Corollary 2
, 
, then for sufficiently small 
inequality (9) and (10) will have the form
Let 
Theorem 6. Let 
be a random field, sample continuous with probability one from the 
and the conditions of Lemma 4 and Lemma 5 hold,
For
, where 
is a monotone increasing continuous function such that 
as
, moreover,
, (11)
where
, and 
is the inverse function to
. Then the function 
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.
Example 1. Let 
be a function such that
, for some 
and all
. Then 
for 
and condition (11) holds for all 
(12)
Condition (12) holds if
, for
,
,
. In this case, the condition of Theorem 6 is satisfied if for 
there exist constants 
such that
, (13)
For 
all
, and sufficiently small
.
Theorem 7. Let 
be a random field, sample continuous with probability one from the space
, where 
is a function such that 
for some 
and all 
and the conditions of Lemma 4 and Lemma 5 hold and 
, 
for some
, 
,
. Then the function 
which is represented in the form (4) is classical solution to the problems (1) and (2).
Proof. It follows from Lemma 5 that there exist integrals with probability one
,
.
According to Theorem 5 to make the function 
be the solution of problems (1) and (2) it is sufficient to prove that integrals (5) and (6) converge uniformly in probability in
, 
to the integrals
, 
for any
,
. According to Theorem 6, using the Example (1), to make integral (5) and (6) converge in probability in 
the following conditions must hold
Using generalized Minkovskoho inequality we obtain
(14)
Let 
and for sufficiently small
, using the inequality (10), we have
(15)
Consider
It follows from Lemma 4 that 
Therefore
(16)
Let 
then
Let 
and for sufficiently small
, using the inequality (9), we have
Therefore
(17)
Thus we obtain from (14), (15), (16) and (17) that
Consider
Since
, we have
Using that
, 
, then the 
and for sufficiently small
, we have
So we have
Therefore 
Then for
, we have
where
, 
are some constants.
Consider
From Lemma 4, we obtain
Similarly 
where
, 
are some constants.
Consider
From Lemma 4 we obtain
Then 
where 
are some constants.
4. Estimates of the Distribution of the Supremum of a Solution
Theorem 8. [9] Let 
be the 
-dimensional space, 
, 
,
. Assume that 
is separable and
. If
where 
is a monotone increasing continuous function such that 
as
, and
, where 
and 
is the inverse function to
. Then
, for all 
and
, where
Theorem 9. Let the conditions of Theorem 6 hold
.
where
. Then
for all 
and
, where
where 
is a monotone increasing continuous function such that 
as
, and 
is the inverse function to
.
Proof. This theorem follows from Theorem 8.