We consider a Cauchy problem for stochastic heat equation driven by a real harmonizable fractional stable process

Partial differential equations with randomness are widely used to model physical, chemical, biological phenomena, financial asset prices, economical processes, etc. The popularity of such models is due to the combination of deterministic and stochastic features among their characteristics.

The majority of existing literature is devoted to the case where the random noise has some Gaussian or sub-Gaussian distribution. To mention only few papers, a heat equation with Gaussian noise was considered in [

The research carried out in the cited articles does not allow one to consider phenomena where the randomness has a heavy-tailed distribution. But heavy tails are ubiquitous when modeling extreme risks, so it is quite important to consider equations with heavy-tailed noise.

The main object of this article is a stochastic heat equation in which the source of randomness is a real harmonizable fractional stable process

We consider approximations for the solution of this equation, which are obtained by truncating the LePage representation series of

The paper is organized as follows. Section 2 contains basic facts about stable random variables and related processes. It also establishes an auxiliary analytical lemma. In Section 3, we formulate and prove the main result of this article.

In this paper, we consider only symmetric

A random variable

To construct families of stable random variables, in particular, stable random processes, one frequently uses some stable random measures. We will be interested in the so-called complex rotationally invariant S

for any Borel set

for any

for any disjoint sets

For a function

Let now

Note that the rotational invariance implies

In the rest of our paper,

We will use the pathwise fractional integration; for more detail, see [

The following result specifies the rate of convergence in the Riemann–Lebesgue lemma. It may be known that, for example, for periodic functions and integer parameter, this is Zygmund’s theorem; however, we failed to find it in the literature. Moreover, a similar reasoning will be used later in the proof of our main results, so we found it suitable to present its proof.

If

Consider a Cauchy problem for the one-dimensional heat equation

We consider Eq. (

Take some

The process

To simplify the following reasoning, we assume that

Let us consider the approximation of the process

For fixed

Taking into account that the processes

Fix arbitrary numbers

Let us transform the differential

Further, recall that

By the strong law of large numbers,

In particular,

It remains to prove that the sum

Now, thanks to the integrability of the fractional derivative of

It is possible to consider (

The authors thank an anonymous referee for his careful reading of the manuscript and valuable remarks, which led to a significant improvement of the article.