SPDEs with α-Stable Lévy Noise: A Random Field Approach. (4th February 2014) - Balan, Raluca M.
Abstract : This paper is dedicated to the study of a nonlinear SPDE on a bounded domain inR d, with zero initial conditions and Dirichlet boundary, driven by anα -stable Lévy noiseZ withα ∈ ( 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 fromZ the jumps which exceed a fixed valueK ), yielding a solutionu K, and then show that the solutionsu L, L > K coincide on the eventt ≤ τ K, for some stopping timesτ K 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 toZ K satisfies ap th moment inequality. This inequality plays the same role as the Burkholder-Davis-Gundy inequality in the theory of integration with respect to continuous martingales.