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

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.