Accessibility of SPDEs driven by pure jump noise and its applications

TL;DR

This paper introduces a novel method to analyze the accessibility of SPDEs driven by pure jump noise, including degenerate cases, and applies it to establish ergodicity for several complex stochastic equations.

Contribution

It develops a new approach for accessibility of SPDEs with degenerate pure jump noise and applies it to key equations like stochastic Navier-Stokes and p-Laplace.

Findings

Established accessibility for a broad class of SPDEs with degenerate pure jump noise.

Proved ergodicity for stochastic p-Laplace and fast diffusion equations driven by Levy processes.

Extended applicability to equations with heavy-tailed jump noise.

Abstract

In this paper, we develop a new method to obtain the accessibility of stochastic partial differential equations driven by additive pure jump noise. An important novelty of this paper is to allow the driving noises to be degenerate. As an application, for the first time, we obtain the accessibility of a class of stochastic equations driven by pure jump degenerate noise, which cover 2D stochastic Navier-Stokes equations, stochastic Burgers type equations, singular stochastic p-Laplace equations, stochastic fast diffusion equations, etc. As a further application, we establish the ergodicity of singular stochastic p-Laplace equations and stochastic fast diffusion equations driven by additive pure jump noise, and we remark that the driving noises could be Levy processes with heavy tails.