The stampacchia maximum principle for stochastic partial differential equations forced by lévy noise

Abstract

In this work, we investigate the existence of positive (martingale and pathwise) solutions of stochastic partial differential equations (SPDEs) driven by a Levy noise. The proof relies on the use of truncation, following the Stampacchia approach to maximum principle. Among the applications, the positivity and boundedness results for the solutions of some biological systems and reaction diffusion equations are provided under suitable hypotheses, as well as some comparison theorems. This article improves the results of [ 15 ] where the authors only considered the case of the Wiener noise; even in this case we improve on [ 15 ] because the coefficients of the principal differential operator are now allowed to depend upon \begin{document}$ t $\end{document} .