Ergodicity for nonlinear stochastic evolution equations with multiplicative Poisson noise

Abstract

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.

Φ-entropy inequality and application for SDEs with jumps

“…Although the existence of invariant probability measures for SDEs with jumps has been investigated in the literature, we did not find any existing result which directly applies to the framework in Theorem 1.1. For instance, in [2,Theorem 4.5] it is assumed that R d |z| 2 ν(dz) < ∞, while in [1] the Lévy process is assumed to be the α-stable process and b(x) is a perturbation by −γx for some constant γ > 0, see also [7,8] for the study of semilinear SPDEs with jump. We aim to present a new result which is sharp in terms of the Lévy measure and, in particular, implies the existence of invariant probability measure in the situation of Theorem 1.1 (2).…”

Ergodicity and Kolmogorov Equations for Dissipative SPDEs with Singular Drift: a Variational Approach

“…For this reason, questions such as ergodicity and existence of invariant measures for (1.1) cannot be addressed using the results by Barbu and Da Prato in [3], which appear to be the only ones available for equations in the variational setting (cf. also [15]). On the other hand, there is a very vast literature on these problems for equations cast in the mild setting, references to which can be found, for instance, in [6,7,17].…”

Ergodicity and Local Limits for Stochastic Local and Nonlocal $p$-Laplace Equations

“…A stochastic variational inequality (SVI) approach to stochastic fast diffusion equations has been developed in [30] and their ergodicity has been considered in [10, 32, 49-51, 66, 67]. The case of stochastic degenerate p-Laplace equations, that is for p > 2, has been investigated in [9,31,47,53,57,[65][66][67] and ergodicity for stochastic porous media equations has been obtained in [8,9,19,20,34,43,47,54,57,64,65].…”