Random field solutions to linear SPDEs driven by symmetric pure jump L\'evy space-time white noises

TL;DR

This paper investigates the existence and equivalence of mild and generalized solutions to linear SPDEs driven by symmetric pure jump Lévy white noise, providing necessary conditions and applying results to classical equations.

Contribution

It introduces conditions for existence and equivalence of solutions to SPDEs driven by jump Lévy noise, including applications to heat, wave, and Poisson equations.

Findings

Identified conditions for solution existence

Established when mild and generalized solutions are equivalent

Provided necessary conditions for random field solutions

Abstract

We study the notions of mild solution and generalized solution to a linear stochastic partial differential equation driven by a pure jump symmetric L\'evy white noise. We identify conditions for existence for these two kinds of solutions, and we identify conditions under which they are essentially equivalent. We establish a necessary condition for the existence of a random field solution to a linear SPDE, and we apply this result to the linear stochastic heat, wave and Poisson equations driven by a symmetric $\alpha$-stable noise.