Accessibility of SPDEs driven by pure jump noise and its applications

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 (possibly degenerate) noise, including stochastic 2D Navier-Stokes equations, stochastic Burgers equations, stochastic singular p p -Laplace equations, and stochastic fast diffusion equations. As a further application, we establish the ergodicity of stochastic singular p p -Laplace equations and stochastic fast diffusion equations driven by additive pure jump noise, and we remark that the driving noises could be Compound Poisson processes or Lévy processes with heavy tails.