Stochastic wave equation in a plane driven by spatial stable noise

The main object of this paper is the planar wave equation \[\bigg(\frac{\partial^2}{\partial t^2}-a^2\varDelta\bigg)U(x,t)=f(x,t),\quad t\ge0, x\in \mathbb {R}^2,\] with random source $f$. The latter is, in certain sense, a symmetric $\alpha$-stable spatial white noise multiplied by some regular function $\sigma$. We define a candidate solution $U$ to the equation via Poisson's formula and prove that the corresponding expression is well defined at each point almost surely, although the exceptional set may depend on the particular point $(x,t)$. We further show that $U$ is H\"{o}lder continuous in time but with probability 1 is unbounded in any neighborhood of each point where $\sigma$ does not vanish. Finally, we prove that $U$ is a generalized solution to the equation.


Introduction
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 [1-3, 5, 6, 8], to mention only a few authors. However, many phenomena are characterized by heavy tails of the corresponding distributions; often not only variances, but also expectations of underlying random variables are infinite. In such cases, the underlying random noise is better modeled by a stable distribution.
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 1 contains the notation and auxiliary information on objects involved. In Section 2, we introduce the main object of the paper, a planar wave equation with stable noise, and establish main results. The existence and spatial properties of a candidate solution to the equation, constructed via Poisson's formula, are studied in Section 2.1. In Section 2.2, we prove that the candidate solution is a generalized solution to the equation. Finally, in Section 2.3, we establish the Hölder regularity of the solution in the time variable.

Notational conventions
Throughout the article, the symbol C denotes a generic constant, the exact value of which is not important and may change from line to line. Similarly, C(ω) is be used to denote a generic a.s. finite random variable. We use the notation |x| both for the absolute value of a real number and for the Euclidean norm of a vector; the particular meaning will be always clear from the context. The Euclidean ball {y : |y − x| ≤ r} is denoted by B(x, r). Finally, R + = [0, +∞).

Stable random variables and measures
In this section, we give essential information on symmetric α-stable SαS random variables and measures; for details, we refer the reader to [7].
Let (Ω, F , P) be a complete probability space. For a number α ∈ (0, 2), called the stability parameter, a random variable ξ is SαS with the scale parameter σ α , σ ≥ 0, if its characteristic function is We also denote ξ α = σ; note that this is a (quasi-)norm for α ≥ 1. SαS random variables and fields are often constructed by means of an independently scattered SαS random measure, which is defined as follows. Denoting by B f (R d ) the family of Borel sets of finite Lebesgue measure, a random set function M : B f (R d ) × Ω → R is called an independently scattered SαS random measure with Lebesgue control measure if 1) for any A ∈ B f (R d ), the random variable M (A) is SαS with scale parameter equal to λ d (A), the Lebesgue measure of A; 2) for any disjoint A 1 , . . . , A n ∈ B f (R d ), the values M (A 1 ), . . . , M (A n ) are independent.

3) for any disjoint
almost surely.
For a function f (x) ∈ L α (R d ), the integral is defined as the limit in probability of integrals of simple compactly supported functions; its value is an SαS random variable with Our analysis is based on the LePage series representation of M defined as follows. Let ϕ be an arbitrary continuous positive probability density function on R d , and {Γ k , k ≥ 1}, {ξ k , k ≥ 1}, {g k , k ≥ 1} be three independent families of random variables satisfying: • Γ k , k ≥ 1, is a sequence of arrivals of a Poisson process with unit intensity; • ξ k , k ≥ 1, are independent random vectors in R d with density ϕ; • g k , k ≥ 1, are independent centered Gaussian variables with E[|g k | α ] = 1.
Then M (as a random process indexed by finite measure Borel sets) has the same finite-dimensional distributions as where C α = Γ (2−α) cos πα Throughout the paper, we work with a planar SαS measure M , that is, we consider the case d = 2. We will assume, without loss of generality, that M is given by (1), so that, for any function f ∈ L α (R 2 ), the integral is given by an almost surely convergent series (2). Moreover, we assume that and, for all ω = (ω Γ , ω ξ , ωg) and k ≥ 1, . This will not harm the generality but will considerably simplify our exposition.

Main results
For a positive constant a > 0, consider the planar wave equation with zero initial conditions. The random source is a product of a continuous function σ and SαS white noiseṀ (x), which is a formal derivative of a planar SαS random measure M introduced in the previous section. The precise meaning of this equality is not immediately obvious. Clearly, there can be no classical (belonging to C 2 (R 2 × R + ) solution to this equation, so we will look at generalized solutions. Let D(R 2 × R + ) denote the class of all compactly supported infinitely continuously differentiable functions on R 2 × R + . By a generalized solution we mean a function satisfying Our approach is to consider a candidate solution given by Poisson's formula and later, in Section 2.2, to show that it solves Eq. (3) in a generalized sense.
The integral in (5) is understood in the following sense: we define In what follows, we need some assumptions about the coefficient σ.

Existence and spatial properties of a candidate solution
First, we establish a result on the existence of the integral defining the candidate solution U (x, t).
Proof. According to the definition of the integral with respect to M , the integral is well defined, provided that Taking into account (S1), we have the estimate Therefore, where ε is a small positive number. This proves the statement.
Recall that M is assumed to coincide with its LePage series, so we have that, for all t ≥ 0 and x ∈ R 2 , U (x, t) is given by the almost surely convergent series We will further see that the exceptional event of zero probability generally depends on x and t. Moreover, if σ is continuous, then U is unbounded in any neighborhood of any point where σ does not vanish. In order to prove this, we first note that G(x, x, t) is infinite for any t ≥ 0 and x ∈ R 2 such that σ(x, t) = 0. Indeed, let σ(x, t) > 0 for some t ≥ 0 and x ∈ R 2 . Then there is The first integral is finite, whereas This observation leads to the following statement.

Generalized solution
Theorem 2 shows that the function U (x, t) cannot be a classical solution to Eq. (3). Our next aim is to show that it solves (3) in a generalized sense.
Remark 4. In the second part of this theorem, the exceptional event of probability zero may depend on θ.
Proof. Write the LePage representation for the left-hand side of Eq. (4): and its right-hand side The proof consists of two steps: showing the convergence of the LePage series and then proving that (4) holds for partial sums of the LePage series. Let us estimate the terms in the series for L(θ). Assume that supp θ ⊂ [0, R] × B(0, R). Then, denoting ψ(x, t) = ∂ 2 ∂t 2 θ(x, t) − a 2 ∆θ(x, t), we have Consider first the case α ∈ (0, 1). By the strong law of large numbers and wellknown properties of Gaussian random variables there exists Ω 0 ∈ F , P(Ω 0 ) = 1, such that, for all ω ∈ Ω 0 and k ≥ 1, where C 1 , C 2 are some positive random variables. Therefore, the kth term in the series for L(θ) is bounded by Consequently, the series for L(θ) is convergent for all ω ∈ Ω 0 and θ(x, t) ∈ D(R 2 × R + ). Similarly, we can show the convergence of R(θ). For α ∈ [1, 2), the argument is changed slightly. Specifically, we show the almost sure convergence of the series for L(θ) and R(θ) for all θ(x, t) ∈ D(R 2 × R + ). Indeed, for fixed ω ξ ∈ Ω ξ , ω Γ ∈ Ω Γ , in view of (9), we have almost surely. Therefore, by the Kolmogorov theorem the series for L(θ) converges P g -almost surely for almost all ω ξ ∈ Ω ξ and ω Γ ∈ Ω Γ and, therefore, P-almost surely. The almost sure convergence of R(θ) is shown in a similar way. Now we prove that Eq. (4) holds for partial sums of the LePage series; the argument does not depend on the value of α. The counterpart of Eq. (4) for the partial sums reads as It suffices to show the equality of the corresponding terms, that is, to prove that, for This equality, in turn, would follow if we show that for all τ ≥ 0, y ∈ R 2 . As before, assume that supp θ ⊂ [0, R] × B(0, R). Define Then Consider the following Cauchy problem: Clearly, the function θ is a solution. On the other hand, by Poisson's formula, for all x ∈ R 2 and r ≥ 0, θ(x, r) = 1 2πa Changing the variables r → R − τ , s → R − t and noticing that ψ(t, x) vanishes for τ ≥ R, we get (10) for all t ∈ [0, R]. For τ ≥ R, the both sides of the equality are zero, whence the proof follows.

Regularity of solution in time variable
In this section, adapting the argument of [4], we show that the solution U constructed by means of Poisson's formula (6) is Hölder continuous in the time variable. Since we have already shown that U is highly irregular in the spatial variable, our findings for the planar wave equation are in a sharp contrast with the scalar case, where the time and space regularity are the same.