Series expansions for SPDEs with symmetric $\alpha$-stable L\'evy noise

TL;DR

This paper develops series expansions for solutions to SPDEs driven by symmetric alpha-stable Levy noise, extending methods beyond Gaussian cases using stable integrals and Levy basis embedding.

Contribution

It provides explicit series solutions for SPDEs with infinite-variance Levy noise, employing stable integrals and Levy basis embedding techniques.

Findings

Derived sufficient conditions for existence of solutions.

Constructed explicit series expansions for solutions.

Applied results to heat and wave equations with heavy-tailed noise.

Abstract

In this article, we examine a stochastic partial differential equation (SPDE) driven by a symmetric $\alpha$-stable (S$\alpha$S) L\'evy noise, that is multiplied by a linear function $\sigma(u)=u$ of the solution. The solution is interpreted in the mild sense. For this models, in the case of the Gaussian noise, the solution has an explicit Wiener chaos expansion, and is studied using tools from Malliavin calculus. These tools cannot be used for an infinite-variance L\'evy noise. In this article, we provide sufficient conditions for the existence of a solution, and we give an explicit series expansion of this solution. To achieve this, we use the multiple stable integrals, which were developed in Samorodnitsky and Taqqu (1990, 1991), and originate from the LePage series representation of the noise. To give a meaning to the stochastic integral which appears in the definition of solution, we embed the space-time L\'evy noise into a L\'evy basis, and use the stochastic integration theory (Bichteler and Jacod 1983, Bichteler 2002) with respect to this object, as in other studies of SPDEs with heavy-tailed noise: Chong (2017a), Chong (2017b), Chong, Dalang and Humeau (2019). As applications, we consider the heat and wave equations with linear multiplicative noise, also called the parabolic/hyperbolic Anderson models.