Stochastic Burgers type equations with two reflecting walls: Existence, uniqueness and exponential ergodicity

Abstract

In this paper, we study stochastic Burgers type equations with two reflecting walls driven by multiplicative noise. We first establish the existence and uniqueness of the solution by Picard iteration technique, the main trouble is to handle the complicated nonlinear term ∂g(u(x,t))∂x which involves the derivative of the solution u(x, t) and the singularities due to reflection. Furthermore, we prove the exponential ergodicity for the equations driven by possibly degenerate, multiplicative noise through asymptotic coupling. More generally, we also give a simplified criterion for exponential ergodicity of parabolic stochastic partial differential equations with reflections driven by degenerate multiplicative noise.