Table of Contents Author Guidelines Submit a Manuscript
International Journal of Stochastic Analysis
Volume 2014, Article ID 793275, 22 pages
http://dx.doi.org/10.1155/2014/793275
Research Article

SPDEs with -Stable Lévy Noise: A Random Field Approach

Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Avenue, Ottawa, ON, Canada K1N 6N5

Received 17 August 2013; Accepted 25 November 2013; Published 4 February 2014

Academic Editor: H. Srivastava

Copyright © 2014 Raluca M. Balan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper is dedicated to the study of a nonlinear SPDE on a bounded domain in , with zero initial conditions and Dirichlet boundary, driven by an -stable Lévy noise with , , 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 the jumps which exceed a fixed value ), yielding a solution , and then show that the solutions coincide on the event , for some stopping times 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 satisfies a 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.