Stochastic heat equation driven by space-only fractional Lévy noise

Abstract

We introduce a novel class of stochastic partial differential equations (SPDEs) driven by space-only fractional Lévy noise. In contrast to the prevalent focus on space-time noise in the existing literature, our work explores the unique challenges and opportunities presented by purely spatial perturbations. We establish the existence and uniqueness of the solution to the stochastic heat equation by rigorously establishing the well-definedness and equivalence of mild and weak solution concepts, utilizing a blend of stochastic, deterministic, and fractional calculus techniques. Specifically, we derive explicit expressions for the covariance and variance functions, and characterize the solution’s law. These results constitute a first step towards a comprehensive understanding of SPDEs with space-only fractional Lévy noise.