Well-posedness and invariant measure for quasilinear parabolic SPDE on a bounded domain

TL;DR

This paper investigates well-posedness, invariant measures, and ergodic properties of quasilinear parabolic SPDEs with multiplicative noise on bounded domains, establishing existence, uniqueness, and stability of solutions.

Contribution

It provides new results on existence, uniqueness, and ergodicity of solutions and invariant measures for a broad class of quasilinear parabolic SPDEs with general noise.

Findings

Existence and uniqueness of solutions in $L^{1}$

Comparison and contraction properties of solutions

Existence and ergodicity of invariant measures

Abstract

We study quasilinear parabolic stochastic partial differential equations with general multiplicative noise on a bounded domain in $\mathbb{R}^{d}$, with homogeneous Dirichlet boundary condition. We establish the existence and uniqueness of solutions in a $L^{1}$ setting, and we prove a comparison result and an $L^{1}$-contraction property for the solutions. In addition, we show the existence of an invariant measure in case of non-degenerate diffusion. Finally, we show the uniqueness and ergodicity of the invariant measure in $L^{1}$, in case of bounded diffusion and additive noise.