Irreducibility and ergodicity of SPDEs driven by pure jump noise
Jian Wang, Hao Yang, Jianliang Zhai, Tusheng Zhang
TL;DR
The authors develop a novel framework to establish irreducibility for SPDEs and SDEs driven by multiplicative pure jump noise under two mild, largely independent assumptions: a weak continuity property for the Markov family ${X^x}$ and a nondegeneracy condition on the Lévy intensity measure $\nu$. Their method bypasses traditional control-theoretic approaches, avoiding approximate controllability, support considerations on path spaces, and Girsanov transforms, and applies to a broad class of equations with locally monotone or singular coefficients, including nonlinear Schrödinger equations and infinite-dimensional compound Poisson noises. This leads to strong irreducibility results and, via the e-property, to ergodicity (uniqueness of invariant measures) for several stochastic evolution inclusions, such as total variation flow, curve shortening flow, and related multi-valued dynamics. The results substantially broaden the scope of irreducibility and ergodicity for systems driven by pure jump noise, enabling analysis of complex SPDEs/SDEs that were previously inaccessible with Gaussian-based or jump-additive techniques.
Abstract
The irreducibility is fundamental for the study of ergodicity of stochastic dynamical systems. The existing methods on the irreducibility of stochastic partial differential equations (SPDEs) and stochastic differential equations (SDEs) driven by pure jump noise are basically along the same lines as that for the Gaussian case, which are not particularly suitable for jump noise. As a result, restrictive conditions are usually placed on the driving jump noise. Basically the driving noises are additive type and more or less in the class of stable processes. In this paper, we develop a new and effective method to obtain the irreducibility of SPDEs and SDEs driven by multiplicative pure jump noise. The conditions placed on the coefficients and the driving noise are very mild, and in some sense they are necessary and sufficient. As an application of our main results, we remove all the restrictive conditions on the driving noises in the literature,and derive new irreducibility results of a large class of equations driven by pure jump noise, including SPDEs with locally monotone coefficients, SPDEs/SDEs with singular coefficients, nonlinear Schrödinger equations, etc. We emphasize that under our setting the driving noises could be compound Poisson processes, even allowed to be infinite dimensional. As further applications of the main results, we obtain the ergodicity of multi-valued, singular stochastic evolution inclusions such as stochastic 1-Laplacian evolution (total variation flow), stochastic sign fast diffusion equation, stochastic minimal surface flow, stochastic curve shortening flow, etc.