Well-posedness and asymptotic behavior for stochastic reaction-diffusion equations with multiplicative Poisson noise

TL;DR

This paper proves well-posedness and investigates the long-term behavior of stochastic reaction-diffusion equations with multiplicative Poisson noise, introducing new inequalities for stochastic convolutions.

Contribution

It establishes well-posedness for a class of stochastic equations with polynomial growth and introduces a novel maximal inequality for stochastic convolutions in $L_p$ spaces.

Findings

Proves well-posedness in the mild sense.

Studies existence and uniqueness of invariant measures.

Develops a new maximal inequality for stochastic convolutions.

Abstract

We establish well-posedness in the mild sense for a class of stochastic semilinear evolution equations with a polynomially growing quasi-monotone nonlinearity and multiplicative Poisson noise. We also study existence and uniqueness of invariant measures for the associated semigroup in the Markovian case. A key role is played by a new maximal inequality for stochastic convolutions in $L_p$ spaces.