Averaging principle for the one-dimensional parabolic equation driven by stochastic measure

A stochastic parabolic equation on [ 0 ,T ] × R driven by a general stochastic measure is considered. The averaging principle for the equation is established. The convergence rate is compared with other results on related topics.

(3) Functions f , σ are defined below.Note that we consider solutions to the formal equations (1) and (3) in the mild form.
Averaging is widely used to describe the asymptotic behaviour of both stochastic and deterministic systems.Stochastic parabolic equation with the random noise represented by a general stochastic measure was introduced in [1].The averaging principle for two-time-scales system driven by two independent Wiener processes was studied, for example, in [7].Some other equations driven by Wiener process are considered in [4,6] and [20].Different types of equations with general stochastic measures are investigated in [2,3,14,19] and [16].
The rest of the paper is organized as follows.Section 2 contains the basic facts concerning stochastic measures and integrals with respect to them.Section 3 contains precise formulation of the problem, assumptions, auxiliary statements while the main result is proved in Section 4. Some examples are given in Section 5.

Preliminaries
Let ( , F, P) be a complete probability space and B be a Borel σ -algebra on R. Denote the set of all real-valued random variables defined on ( , F, P) as L 0 = L 0 ( , F, P), convergence in L 0 means the convergence in probability.
In other words, μ is a vector measure with values in L 0 .We do not assume any martingale properties or moment existence for SM.
Consider some examples of SMs.If M t is a square integrable martingale then Some other examples can be found in [14].
In [10,Chapter 7] the definition of the integral A g dμ, where g : R → R is a deterministic measurable function, A ∈ B and μ is an SM, is given and its properties are studied.In particular, every bounded measurable g is integrable with respect to (w.r.t.) any μ.This integral was constructed and studied in [10] for μ defined on an arbitrary σ -algebra, but in our paper, we consider SM on Borel subsets of R.
In the sequel, μ denotes a SM, C and C(ω) denote positive constant and positive random constant, respectively, whose exact values are not important (C < ∞, C(ω) < ∞ a.s.).
We will use the following statement.
Lemma 1 (Lemma 3.1 in [12]).Let φ l : R → R, l ≥ 1, be measurable functions such that φ We consider the Besov spaces B α 22 ([c, d]), 0 < α < 1, with a standard norm , where For any j ∈ Z and all n ≥ 0, put The following lemma is a key tool for estimates of the stochastic integral.Lemma 2 (Lemma 3 in [13]).Let Z be an arbitrary set, and the function q(z, s) : Then the random function Theorem 1.1 [9] implies that for α = (β + 1)/2, From Lemma 1 it follows that for each

Formulation of the problem and auxiliary lemmas
We consider the mild solutions to (1), i.e. the measurable random functions Here p is the fundamental solution of operator L from (2).We will refer to the following assumptions on f , σ , u 0 . where for some constants L σ , 1/2 < β(σ ) < 1, and all s ∈ R + , y 1 , y 2 ∈ R.
Assumption E4.There exist the limits Note that if f , σ satisfy the conditions E2 and E3, respectively, f and σ satisfy them as well.We will show that for f ; the proof for σ is analogous.f is measurable as a limit of measurable functions, its boundedness is obvious while Lipschitz condition follows from the inequalities Therefore, functions Assertion E5 holds, for example, if f (s, y, v) and σ (s, y) are periodic in s for each y, v, and the set of values of minimal period is bounded.
Assumption L. Functions a(t), b(t), c(t) are continuous in [0, T ], and for some positive constants β, L, δ the following inequalities hold in [8, section 4, Theorem 1] shows that under assumption L the fundamental solution exists and where (without loss of generality we can say that constant M in ( 12) is the same as in ( 8)-( 11)).Let the stochastic measure μ satisfy the following condition.
The following lemma is an analogue of [15, Lemma 3].

The main result
We are ready to formulate the main result of the paper.