The main object of this paper is the planar wave equation

Stochastic partial differential equations are widely used in modeling different phenomena involving randomness, and the area of their application is constantly increasing. This is reflected by increasing number of works devoted to such equations. Vast majority of these articles is devoted to the case where the underlying noise is Gaussian. In particular, a stochastic wave equation with Gaussian noise was studied in [

In this paper, we study a wave equation in the plane, where the random source has a stable distribution. We prove that a candidate solution to the equation, constructed by means of Poisson’s formula, is a generalized solution. We also show that it is Hölder continuous is time variable, but it is irregular in the spatial variable.

The paper is organized as follows. Section

Throughout the article, the symbol

In this section, we give essential information on symmetric

Let

S

for any

for any disjoint

for any disjoint

Our analysis is based on the LePage series representation of

Then

For a positive constant

Let

Our approach is to consider a candidate solution given by Poisson’s formula

The integral in (

In what follows, we need some assumptions about the coefficient

Boundedness:

Continuity:

Hölder continuity in time: there exists

First, we establish a result on the existence of the integral defining the candidate solution

According to the definition of the integral with respect to

Recall that

We will further see that the exceptional event of zero probability generally depends on

Define

Since

Consequently, for all

Theorem

In the second part of this theorem, the exceptional event of probability zero may depend on

Write the LePage representation for the left-hand side of Eq. (

Let us estimate the terms in the series for

Consider first the case

For

Now we prove that Eq. (

As before, assume that

In this section, adapting the argument of [

Take some