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.

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

The averaging principle for fractional differential equations driven by Lévy noise is established by Shen et al. [

The averaging principle for equations driven by general stochastic measures is considered in [

Our aim here is to establish the averaging principle for the cable equation driven by stochastic measure studied in [

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

Let

Consider the mild solution to the following equation:

Let

We study the convergence

The rest of the paper is organized as follows. Section

To prove the convergence of solutions, we will apply an estimate of a stochastic integral using the norm of the Besov space

Denote

Let

By [

According to [

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

Condition 1. The function

Condition 2. The function

By [

The proof of Theorem 5.1 ([

The following assumptions are used in the rest of the paper.

A1. The function

A2.

A3. Limit (

The reasoning is the same as that used in [

Step 1. By obtaining the Hölder condition for the stochastic integral with respect to the space variable we get the analogues of [

For any fixed

Consider

On the other hand,

Raise the inequality (

For the same reason,

Further, we can repeat the proof of Theorem 3.1 ([

Besides, as mentioned above, we use [

Step 2. The reasoning is the same in the case of the Hölder property with respect to the time variable. Given

Moreover, denote

Now we consider the case of

We use the change of variables

Similarly to (

Therefore,

Then, by assumption A2, we get

On the other hand, the same reasoning as that used in obtaining bound (

Now we raise the inequality (

Consequently, taking to consideration (

From this point we can repeat the proof of Theorem 4.1 ([

Therefore, we obtain the Hölder condition with respect to the time variable of the order

Step 3. The last part of the proof is analogous to the proof of Theorem 5.1 in [

Then we use the change of variables

To prove the Hölder regularity with respect to the time variable we consider

According to estimates (

The similar arguments yield

Finally,

In consequence,

Likewise, we get the same estimate for

The rest of the proof runs as respective part of the proof of Theorem 5.1 in [

In this section we consider the random functions

By Theorem

Let

Then

Denote

In the integral from (

We have

Now we estimate

Since

Analogously,

Substitute obtained bounds to (

Hence,

Next we estimate the integrals from the last relation. For

Therefore,

Consequently,

In the same way we obtain

Therefore, we get

Now we estimate

By condition A3 and finiteness of

Denote

Since

For

Consider the second integral. We have

According to (

Hence,

Then we use the same reasoning as in getting (

It follows that

The same estimate holds for term

Taking into account (

Raise the latter inequality to the power

Thus, for any

Consequently, by relation (

The author is grateful to the anonymous referee for careful reading of the paper and valuable comments and suggestions.