Lévy driven linear and semilinear stochastic partial differential equations

Abstract

The goal of this paper is twofold. In the first part we will study Lévy white noise in different distributional spaces and solve equations of the type $p(D)s=q(D)\dot{L}$, where $p$ and $q$ are polynomials. Furthermore, we will study measurability of $s$ in Besov spaces. By using this result we will prove that stochastic partial differential equations of the form \begin{align*} p(D)u=g(\cdot,u)+\dot{L} \end{align*} have measurable solutions in weighted Besov spaces, where $p(D)$ is a partial differential operator in a certain class, $g:{\mathbb{R}}^d\times \mathbb{C}\to \mathbb{R}$ satisfies some Lipschitz condition and $\dot{L}$ is a Lévy white noise.