Ergodicity for nonlinear stochastic evolution equations with multiplicative Poisson noise

TL;DR

This paper investigates the long-term behavior of nonlinear stochastic evolution equations with jump noise, establishing conditions for ergodicity, invariant measures, and solvability of related Kolmogorov equations in a variational framework.

Contribution

It provides new sufficient conditions for existence, ergodicity, and uniqueness of invariant measures for a broad class of nonlinear PDEs with multiplicative Poisson noise.

Findings

Established ergodicity and invariant measures for nonlinear stochastic equations

Proved solvability of the Kolmogorov equation in $L_1$ spaces

Covered a wide class of PDEs with jump noise in the variational setting

Abstract

We study the asymptotic behavior of solutions to stochastic evolution equations with monotone drift and multiplicative Poisson noise in the variational setting, thus covering a large class of (fully) nonlinear partial differential equations perturbed by jump noise. In particular, we provide sufficient conditions for the existence, ergodicity, and uniqueness of invariant measures. Furthermore, under mild additional assumptions, we prove that the Kolmogorov equation associated to the stochastic equation with additive noise is solvable in $L_1$ spaces with respect to an invariant measure.