SPDEs with α-Stable Lévy Noise: A Random Field Approach

Abstract

This paper is dedicated to the study of a nonlinear SPDE on a bounded domain in Rd, with zero initial conditions and Dirichlet boundary, driven by an α-stable Lévy noise Z with α∈(0,2), α≠1, and possibly nonsymmetric tails. To give a meaning to the concept of solution, we develop a theory of stochastic integration with respect to this noise. The idea is to first solve the equation with “truncated” noise (obtained by removing from Z the jumps which exceed a fixed value K), yielding a solution uK, and then show that the solutions uL,L>K coincide on the event t≤τK, for some stopping times τK converging to infinity. A similar idea was used in the setting of Hilbert-space valued processes. A major step is to show that the stochastic integral with respect to ZK satisfies a pth moment inequality. This inequality plays the same role as the Burkholder-Davis-Gundy inequality in the theory of integration with respect to continuous martingales.