Averaging principle for a stochastic cable equation

We consider the cable equation in the mild form driven by a general stochastic measure. The averaging principle for the equation is established. The rate of convergence is estimated. The regularity of the mild solution is also studied. The orders in time and space variables in the Holder condition for the solution are improved in comparison with previous results in the literature on this topic.


Introduction
Averaging methods are important for describing and investigating the asymptotic behavior of dynamical systems. Therefore, the theory of the averaging principle for stochastic differential equations is a fascinating modern topic and many mathematicians work quite actively within this field. For instance, a weak order in averaging for wave equations with L 2 -valued Wiener processes is studied in [14]. Bao et al. [3] considered two-time-scale equations with α-stable noises. Strong and weak orders in averaging for stochastic partial differential equations with Wiener processes are given in [10,11,13].
The averaging principle for fractional differential equations driven by Lévy noise is established by Shen et al. [31]. Wang and Xu [34] investigated the stochastic av-eraging method for neutral stochastic delay equations driven by fractional Brownian motion with Hurst parameter H ∈ (1/2, 1). Other interesting examples of studying of averaging for stochastic differential equations can be found in papers [1,12,15,18].
Our aim here is to establish the averaging principle for the cable equation driven by stochastic measure studied in [25]. For this purpose we also improve the orders of the Hölder condition for the mild solution with respect to space and time variables obtained in [25,Theorem 5.1] (see Theorem 1 below).
Properties of the mild solutions to stochastic partial differential equations are studied in a number of papers. In particular, the existence and uniqueness of a solution for the class of non-autonomous parabolic stochastic partial differential equations defined on a bounded open subset D ⊂ R d and driven by an L 2 (D)-valued fractional Brownian motion with the Hurst index H > 1/2 are proved in [28]. In [2] the ergodic property of the solution to a fractional stochastic heat equation is established. Wave equations with general stochastic measures and α-stable distributions are investigated in the papers [5,7,8,24] and [20,29,19], respectively.
Let L 0 ( , F, P) be the set of all real-valued random variables defined on a complete probability space ( , F, P), X be an arbitrary set and B(X) be a σ -algebra of Borel subsets of X. Let μ be a stochastic measure on B(X), i.e. a σ -additive mapping μ : B(X) → L 0 ( , F, P). Such μ is also called a general stochastic measure (see, for example, [17,Section 7]). Examples of stochastic measures can be found in [17,21,30].
Consider the mild solution to the following equation: where (t, x) ∈ [0, T ] × [0, L], T > 0, L > 0, ε > 0, and μ is a stochastic measure defined on the Borel σ -algebra B([0, L]). Let G be the fundamental solution of the homogeneous cable equation, that is We study the convergence is the mild solution of the averaged equation, that is, The rest of the paper is organized as follows. Section 2 contains some basic facts concerning the estimates of stochastic integrals with respect to general stochastic measures. In Section 3, we study the Hölder regularity of the mild solution of the cable equation with respect to the set of all variables. The averaging principle for the cable equation is established in Section 4.

Preliminaries
To prove the convergence of solutions, we will apply an estimate of a stochastic integral using the norm of the Besov space is finite. Here  such that, for all ε > 0, ω ∈ , z ∈ Z, This version is the same for all z ∈ Z. According to [ where α = ε/2 + 1/2, the constant C depends on α, t and does not depend on z, ω.
Here and in what follows the same symbol C denotes some positive constants that may be different in different places of the paper. The precise values of these constants are not important for our purposes.

Regularity of the mild solution of a cable equation
Regularity of the mild solution (8) of a cable equation driven by a general stochastic measure is studied in [25]. It was proved there that the paths of the solution are Hölder continuous. The following conditions was considered. Condition 1. The function σ (s, y) : [0, T ] × [0, L] → R has the derivative ∂ 2 σ ∂t∂x , which is continuous with respect to the pair of arguments. Condition 2. The function u 0 (y) = u 0 (y, ω) : [0, L] × → R is measurable and has the derivative ∂u 0 ∂y , which is continuous with respect to y and bounded for all fixed ω ∈ .
The following assumptions are used in the rest of the paper. A1. The function u 0 (y) = u 0 (y, ω) : [0, L] × → R is measurable and Hölder continuous, that is
Proof. The reasoning is the same as that used in [25] with some differences. We have divided the proof into 3 steps: the Hölder continuity of the stochastic integral with respect to the space variable; the Hölder continuity of the stochastic integral with respect to the time variable and the Hölder continuity of the function u(t, x).
Step 1. By obtaining the Hölder condition for the stochastic integral with respect to the space variable we get the analogues of [25, inequalities (3.1) and (3.2)]. We use Assumption A2 instead of Condition 1.
For any fixed where notation is used (see [25]).
Consider n = 0, 1. By A2, we have For details see the method used for obtaining bound (35) below. The same estimation holds for term F 1 . Therefore, On the other hand, Since the function σ is bounded and Hölder continuous (by Assumption A2), and The same estimate holds for term F 2 . Hence, Raise the inequality (12) to the power θ 1 and multiply by inequality (11) raised to the power 1 − θ 1 , for an arbitrary θ 1 ∈ (0, 1). We have for all n = 0. For the same reason, for all n = 0, 1.
Step 2. The reasoning is the same in the case of the Hölder property with respect to the time variable. Given x and t 1 < t 2 use the notation of paper [25], that is Moreover, denotē

4(t−s)
. Now we consider the case of n = 0, 1, since the case of n = 0, 1 is the same as in the [25] with reference to [4, Lemma 1 and Lemma 2] instead of [21, Lemma 5.1 and Lemma 6.1] respectively. Namely, we get Hölder continuity of the order 1/4 for these terms.
We use the change of variables s → s + t 2 − t 1 in the first integral of (15) and obtainḡ Similarly to (12), by A2, we have and Therefore, where we used the estimate for s in the domain of integration.
Then, by assumption A2, we get On the other hand, the same reasoning as that used in obtaining bound (12) proves Now we raise the inequality (19) to the power θ 2 and multiply by inequality (18) raised to the power 1 − θ 2 , for an arbitrary θ 2 ∈ (0, 1). Thus we see that Consequently, taking to consideration (17), From this point we can repeat the proof of Theorem 4.1 ( [25]). Instead of using (4.2) of [25], we use bounds (20) and (21) above. Since the integral from (6) is finite for θ 2 β(σ ) > 1/2, Therefore, we obtain the Hölder condition with respect to the time variable of the order where we also count the case n = 0, 1.
Step 3. The last part of the proof is analogous to the proof of Theorem 5.1 in [25]. The integral L 0 G(t, x, y)u 0 (y) dy satisfies condition (10). To prove the Hölder regularity with respect to the space variable we consider x 1 < x 2 and denote Then we use the change of variables y → y + x 2 − x 1 and y → y − x 2 + x 1 in the integrals involving x 2 and G − (t, x, y), G + (t, x, y) respectively. Thus where we use the boundedness of the function u 0 due to its Hölder continuity on To prove the Hölder regularity with respect to the time variable we consider δ ≤ t 1 < t 2 ≤ T . We get According to estimates (34), (25) and (27) from Section 4, The similar arguments yield where we use bound (30) (see Section 4 below). Finally, In consequence, Likewise, we get the same estimate for G + (t 1 , x, y The rest of the proof runs as respective part of the proof of Theorem 5.1 in [25] with the use of Steps 1 and 2 instead of [25, Theorems 3.1 and 4.1].

Averaging principle
In this section we consider the random functions u ε andū given by equations (3) and (4).
Next we estimate the integrals from the last relation. For |y| ≤ L and |x| ≤ L it is clear that Therefore, where we use the relations Consequently, In the same way we obtain which yields Therefore, we get Now we estimate |g(t, x, y + h) − g(t, x, y)| and then w 2 2,[0,L] (g, r) in terms of some positive power of h. Thus, By condition A3 and finiteness of ∞ n=−∞ e − D n± s , we obtain Denote Recall that here h ∈ [0, L] and we consider the integrals in (32) such that y + h ≤ L. So Since e −(t−s) ≤ 1, for t − s ≥ 0, we conclude by Assumption A2 that According to (28), and analogously to (25) we obtain e − The same estimate holds for term |I 1+ | ≤ Ch β(σ ) . Hence, Taking into account (33), we deduce that Raise the latter inequality to the power θ and multiply by inequality (31) raised to the power 1 − θ , for an arbitrary θ ∈ (0, 1). We have w 2 2,[0,L] (g, r) ≤ Cr 2θβ(σ ) ε 1−θ .