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 ![](https://www.scirp.org/html/htmlimages\16-7401845x\a63d19f4-ea9a-4bc4-ae0a-2b75d57a4b96.png)
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 ![](https://www.scirp.org/html/htmlimages\16-7401845x\da1339e6-7b85-448a-bf80-2c2c07bcca05.png)
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
![](https://www.scirp.org/html/htmlimages\16-7401845x\97d31f08-1def-4994-9373-1f2b5b34375c.png)
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
![](https://www.scirp.org/html/htmlimages\16-7401845x\0fbe97b1-4ba4-4c4b-91c2-f37d288c78c4.png)
exist. Let there exist a monotone increasing continuous function
,
, such that
as
for
,
,
and
,
. Assume that
![](https://www.scirp.org/html/htmlimages\16-7401845x\8d16dbc2-852f-4979-843d-0147870cdb8b.png)
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
![](https://www.scirp.org/html/htmlimages\16-7401845x\90fadc8c-4eb6-4c67-aca6-ce3a58386b57.png)
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
![](https://www.scirp.org/html/htmlimages\16-7401845x\4ed9c8c2-2537-48cf-a523-a61ef0b126ba.png)
exist, and the inverse integral Fourier transform
![](https://www.scirp.org/html/htmlimages\16-7401845x\35384667-7147-4a1f-a7bd-3a81ca53085e.png)
exist.
Theorem 5. Let the conditions of Lemma 4 be satisfied and
![](https://www.scirp.org/html/htmlimages\16-7401845x\77c6a315-cd82-4533-92c4-7f57b15cd76b.png)
![](https://www.scirp.org/html/htmlimages\16-7401845x\08e1c47d-829e-49be-ab96-6761dd60e934.png)
and
(4)
If the following integrals exist
,
,
![](https://www.scirp.org/html/htmlimages\16-7401845x\58bd7d75-96a7-48ef-8862-fe0be1731a6e.png)
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
![](https://www.scirp.org/html/htmlimages\16-7401845x\d9548b10-82de-42b4-b8bc-0707eb3763fb.png)
converges with probability one to
![](https://www.scirp.org/html/htmlimages\16-7401845x\11d1f0a8-ca25-41f7-822b-1c3644641b01.png)
uniformly for
,
, Let
(7)
By deriving (7) with respect to
and
, we easily see that
![](https://www.scirp.org/html/htmlimages\16-7401845x\215d4246-25a0-46ab-b423-6aa67e2fbc41.png)
![](https://www.scirp.org/html/htmlimages\16-7401845x\72733185-1012-4d87-9f22-a6e6dee9f8a3.png)
![](https://www.scirp.org/html/htmlimages\16-7401845x\fee67930-c6e1-413c-8ada-5777260c3721.png)
Since for
converges to
, and
converges to
uniformly for
,
with probability one, we conclude that
satisfies Equation (1).
Indeed, ![](https://www.scirp.org/html/htmlimages\16-7401845x\88ffd18e-3291-4572-b851-6099e9872fd6.png)
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 ![](https://www.scirp.org/html/htmlimages\16-7401845x\cce7e7c7-26c0-4957-adeb-4f4de2805f4e.png)
3.
, at
or
.
Then Lebesgue integrals
![](https://www.scirp.org/html/htmlimages\16-7401845x\934c4cb9-78af-41c2-bded-dcd0f19101ee.png)
![](https://www.scirp.org/html/htmlimages\16-7401845x\5a0a6d82-837d-4060-845e-7ff6a30091f4.png)
exist with probability one.
Proof. We shall prove the existence of the integral
![](https://www.scirp.org/html/htmlimages\16-7401845x\d587cbfb-fc0a-4d6f-8cb8-9d3b4dfc7228.png)
For existence of this integral with probability one it is enough to prove that there exists following integral
![](https://www.scirp.org/html/htmlimages\16-7401845x\c38bd865-8292-4f33-a933-d962d055d960.png)
There is an inequality
![](https://www.scirp.org/html/htmlimages\16-7401845x\b1cd4146-7db3-4a4d-ba03-9d150d59642d.png)
Consider
![](https://www.scirp.org/html/htmlimages\16-7401845x\03efafdf-c4f5-4670-867a-1f8244dc5450.png)
Integrating by parts and using the conditions of the lemma, we obtain for y ≠ 0
![](https://www.scirp.org/html/htmlimages\16-7401845x\e22857c8-471f-4043-b2bf-5385acfebb59.png)
![](https://www.scirp.org/html/htmlimages\16-7401845x\e7a08829-4670-453f-b047-3cc262d86480.png)
Then
![](https://www.scirp.org/html/htmlimages\16-7401845x\0c32490a-0810-4515-9561-1e0a3beb95d8.png)
Therefore
![](https://www.scirp.org/html/htmlimages\16-7401845x\d770fae9-002e-40ba-bda9-827039f135d5.png)
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. ![](https://www.scirp.org/html/htmlimages\16-7401845x\4cd7e808-f76c-48ed-a1f6-09b1c7dfb56e.png)
2.
for all
,
. Let
,
be a continuous increasing function such that
for all
, and the function
is increasing for
, and for some constant
.
Then ![](https://www.scirp.org/html/htmlimages\16-7401845x\c5618060-83e6-4c66-b769-83e42d9727d3.png)
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
![](https://www.scirp.org/html/htmlimages\16-7401845x\fcc7e1ff-e408-4ce8-9470-01756d2df0ae.png)
![](https://www.scirp.org/html/htmlimages\16-7401845x\707e1e72-e496-4f41-a613-19056855b535.png)
Let
![](https://www.scirp.org/html/htmlimages\16-7401845x\48c3089b-80c5-45fc-ac99-5abaae81c349.png)
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 ![](https://www.scirp.org/html/htmlimages\16-7401845x\25615379-8643-46b5-a1eb-515525079b51.png)
(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
![](https://www.scirp.org/html/htmlimages\16-7401845x\d1deb2f8-e32a-414b-a0ae-e36c8dea879c.png)
Using generalized Minkovskoho inequality we obtain
(14)
Let
and for sufficiently small
, using the inequality (10), we have
(15)
Consider
![](https://www.scirp.org/html/htmlimages\16-7401845x\b44b6989-609e-41ab-89e8-e2a2176e1942.png)
It follows from Lemma 4 that ![](https://www.scirp.org/html/htmlimages\16-7401845x\4bc047b2-e4aa-43c1-9ae0-db21f5bb6133.png)
Therefore
(16)
Let
then
![](https://www.scirp.org/html/htmlimages\16-7401845x\34faa6b0-10b9-44b4-bc75-7350d9e513f8.png)
Let
and for sufficiently small
, using the inequality (9), we have
![](https://www.scirp.org/html/htmlimages\16-7401845x\97e8d2d7-2b23-4c60-b950-5d40fc05e750.png)
Therefore
(17)
Thus we obtain from (14), (15), (16) and (17) that
![](https://www.scirp.org/html/htmlimages\16-7401845x\24008a51-b6ad-44b7-8a76-85955be39fcb.png)
Consider
![](https://www.scirp.org/html/htmlimages\16-7401845x\c6006258-ec06-4f42-9ff8-ee1631934873.png)
Since
, we have
![](https://www.scirp.org/html/htmlimages\16-7401845x\0f9ee8dc-9742-4824-bf49-947146f8362e.png)
![](https://www.scirp.org/html/htmlimages\16-7401845x\6b6fa58c-99c0-43dc-81cc-b8fa363d32d0.png)
Using that
,
, then the
and for sufficiently small
, we have
![](https://www.scirp.org/html/htmlimages\16-7401845x\57a0601e-c582-4e3b-bdd6-faf0ecb155b5.png)
So we have
![](https://www.scirp.org/html/htmlimages\16-7401845x\70b3b36e-f4cc-4f2f-93c8-c1e2a177b81b.png)
![](https://www.scirp.org/html/htmlimages\16-7401845x\33ca1c0a-8f6c-4f3a-8353-d8289ef9affd.png)
![](https://www.scirp.org/html/htmlimages\16-7401845x\14d044b4-9925-45a7-8196-226431489ebc.png)
![](https://www.scirp.org/html/htmlimages\16-7401845x\d78a6372-cf34-460a-a06f-92d8cc474971.png)
![](https://www.scirp.org/html/htmlimages\16-7401845x\3b5b8597-7b19-4f1d-b50f-7692365c02da.png)
Therefore
Then for
, we have
where
,
are some constants.
Consider
![](https://www.scirp.org/html/htmlimages\16-7401845x\95bc2813-4882-43bc-9d69-6431826cc981.png)
![](https://www.scirp.org/html/htmlimages\16-7401845x\784f480f-b27e-445c-9ea5-c5cbd75104a8.png)
From Lemma 4, we obtain
![](https://www.scirp.org/html/htmlimages\16-7401845x\a0658be6-ef31-4fb1-a59a-1162b89dc35a.png)
Similarly ![](https://www.scirp.org/html/htmlimages\16-7401845x\239b4a46-1782-49a3-827c-e01c6f5939e1.png)
where
,
are some constants.
Consider
![](https://www.scirp.org/html/htmlimages\16-7401845x\68f463da-f349-4cc4-9722-cdbc262d0815.png)
![](https://www.scirp.org/html/htmlimages\16-7401845x\353efd46-7f70-4963-bf22-6364e356b04b.png)
From Lemma 4 we obtain
![](https://www.scirp.org/html/htmlimages\16-7401845x\a61bee6b-9e65-4aac-a558-f9a151e4b3c1.png)
Then ![](https://www.scirp.org/html/htmlimages\16-7401845x\02bcb1c4-b76f-437e-8054-e53de867839f.png)
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
![](https://www.scirp.org/html/htmlimages\16-7401845x\ded1ba17-f256-4fad-a9ba-d701df7e29f7.png)
![](https://www.scirp.org/html/htmlimages\16-7401845x\89c2e2a2-6386-447c-9846-1c523e9647ce.png)
![](https://www.scirp.org/html/htmlimages\16-7401845x\ff36b7d8-66df-43bc-aab6-fc4a15e61891.png)
Theorem 9. Let the conditions of Theorem 6 hold
.
where
![](https://www.scirp.org/html/htmlimages\16-7401845x\77fa32ae-70d4-49f8-89c7-2dbb6ab1b0ce.png)
![](https://www.scirp.org/html/htmlimages\16-7401845x\f2f4ee12-d7a0-4ad2-9362-917691eef9c3.png)
. Then
![](https://www.scirp.org/html/htmlimages\16-7401845x\2857a9d8-6208-4f66-8811-6339725364a6.png)
for all
and
, where
![](https://www.scirp.org/html/htmlimages\16-7401845x\671293b0-27ea-4029-9c11-e6775e502956.png)
![](https://www.scirp.org/html/htmlimages\16-7401845x\6ff6ccbd-921f-4c84-ac0f-658ffe0ccb52.png)
![](https://www.scirp.org/html/htmlimages\16-7401845x\28b125fe-5f4b-4dee-a4ad-74aba4d0399e.png)
where
is a monotone increasing continuous function such that
as
, and
is the inverse function to
.
Proof. This theorem follows from Theorem 8.