Random field solutions to linear SPDEs driven by symmetric pure jump Lévy space-time white noises

Abstract

We study the notions of mild solution and generalized solution to a linear stochastic partial differential equation driven by a pure jump symmetric Lévy white noise, with symmetric $\alpha $-stable Lévy white noise as an important special case. We identify conditions for existence of these two kinds of solutions, and, together with a new stochastic Fubini theorem, we provide conditions under which they are essentially equivalent. We apply these results to the linear stochastic heat, wave and Poisson equations driven by a symmetric $\alpha $-stable Lévy white noise.