MEAN PERIODIC SOLUTIONS OF A INHOMOGENEOUS HEAT EQUATION WITH RANDOM COEFFICIENTS

. We present conditions ensuring the periodicity of the mathematical expectation of a solution of a scalar linear inhomogeneous heat equation with random coeﬃcients where the coeﬃcient in front of the unknown functions is Gaussian or it is uniformly distributed. The obtained results may be treated as ﬁnding a control ensuring the periodicity of the mathematical expectation of a solution of the heat equation.


Introduction. Existence conditions of periodic solutions of equation
with deterministic periodic functions ε and f are well known [7]. The problem becomes much more complicated if ε and f are random processes. The stability of solutions of differential equations with random right-hand side was studied, e.g., in [3].
In [9], [10], the conditions are given under which the mathematical expectation of a solution of equation (1) is periodic.
Consider the Cauchy problem where ε(t) and f (t, x) are independent random processes given by the characteristic functionals ϕ ε (u) and ϕ f (v) (the dependence on random events is not indicated).
A solution of problem (2), (3) is a random process. It is said to be mean periodic with respect to t if its mathematical expectation is a periodic function with respect to t.
In this paper we will study the mean periodicity of a solution of equation (2) with respect to t ∈ R + for two cases.
If a(s) ≡ 0, then we assume that 2. Auxiliary information. Further the notion of variational derivative is used. We give the corresponding definition. Let L 1 (R + ) be the space of complex-valued functions integrable on R + equipped with the norm R+ |u(s)|ds, ϕ be a functional on the space L 1 (R + ), and let h ∈ L 1 (R + ). If where the integral is treated in the Lebesgue sense and is a linear functional bounded with respect to h, then ψ(t, u) is referred to as the variational derivative of the functional ϕ at the point u.
We will use the function χ(s, t, τ ) = χ(s, t)(τ ), defined on R + as follows: χ(s, t, τ ) is equal to sign(τ − s) for τ belonging to the closed interval with endpoints s and t and to zero for other τ .
where the semi-group U x (t) is defined by the relation ν ∈ R is a parameter.
As usual, I stands for the identity operator.
3. Operator W (t, s) and its properties. At first we give here a some explanation concerning the inverse operator U −1 x (t). The semi-group U x (t) transfers the initial value y(0, x) = y 0 (x) into a solution of the initial value problem It should be noted (see, for instance, [6], p. 196) the uniqueness of a continuous bounded on the whole range of variables solution of problem (7). For fixed t = t 0 the function y(t 0 , x) is called as the realization of solution at t = t 0 . The operator U −1 x (t 0 ) transfers the realization y(t 0 , x) into the corresponding initial value y 0 (x). Thus, the domain of the linear operator U −1 x (t) with fixed t is a set of realizations of solutions of equation (7) and the range of values of this operator is a set of initial values {y 0 (x)}.
Let C(L 1 (R + )) be the space of continuous bounded functionals on Introduce the operator W (t, s) by the following way Lemma 3.1. The operator W (t, s) has the following properties: Proof. Using the definition of the operator W (t, s), we get (8), namely

Further we have
From here (9) follows.
We set in (9) at first s = t, τ = s and then t = s, τ = t. Taking into account (8) we get

GALINA KURINA AND VLADIMIR ZADOROZHNIY
The property (10) follows from the two last equalities.
Proof. Let ε(·) have characteristic functional (4). We will use the properties of the semi-group U x (t) and the equality t+ω t a(s)ds = ω 0 a(s)ds which is correct for each t ∈ R and any ω−periodic function a(·). Taking into account the symmetry of the function b(s 1 , s 2 ) and the Fubini theorem we have Next, using the properties of the semi-group U x (t), we have This coincides with the above-obtained expression. Thus the first assertion (11) is proved.
The second assertion (12) can be proved in a similar way. Let ε(·) have characteristic functional (5), then in view of (5) and properties of the semi-group U x (t) we have By comparing the right-hand sides above-represented results, we obtain the first property (11). The second property (12) can be proved in a similar way.
4. Periodicity of mathematical expectation. is the ω-periodic with respect to t ∈ R + mathematical expectation of a solution of equation (2).
Proof. We rewrite formula (6) with the use of the operator W(t, s) in the form If E(y(t, x)) is an ω-periodic function of t, then E(y(0, x)) = y 0 (x) = E(y(ω, x)).
We write out the last relation applying (14) From here, we obtain the following equation for the initial condition: In view of the assumptions of the theorem, (I − W (ω, 0)) −1 exists, therefore, By substituting this expression into relation (14) and by taking into account the properties of the operator W (t, s), we obtain the representation Substituting this expression into relation (15) and combining integrals, we obtain the representation (13) for E(y(t, x)). Let us prove that E(y(t, x)) from (13) is an ω-periodic function of t. It is not difficult to verify that W (τ, 0) commutes with (W (0, ω) − I) −1 . For all t ∈ R + , using the ω-periodicity with respect to t of E(f (t, x)) and properties of the operator W (t, s), we have y(t, x)).
The proof of the theorem is complete. Remark 1. The invertibility of the operator I − W (ω, 0) implies that the homogeneous equation corresponding to (2) has no non-zero mean ω-periodic solutions. is the ω-periodic with respect to t ∈ R + mathematical expectation of a solution of equation (2). is the ω-periodic with respect to t ∈ R + mathematical expectation of a solution of equation (2).

Remark 2.
We can consider the function f (t, x) in (2) as a control function. Then the obtained results may be treated as finding a control ensuring the periodicity of the mathematical expectation of a solution of heat equation (2).