Exponential mixing for the white-forced damped nonlinear wave equation

Abstract

The paper is devoted to studying the stochastic nonlinear wave (NLW) equation$$\partial_t^2 u + \gamma \partial_t u - \triangle u + f(u)=h(x)+\eta(t,x)$$in a bounded domain $D\subset\mathbb{R}^3$. The equation is supplemented with the Dirichlet boundary condition. Here $f$ is a nonlinear term, $h(x)$ is a function in $H^1_0(D)$ and $\eta(t,x)$ is a non-degenerate white noise. We show that the Markov process associated with the flow $\xi_u(t)=[u(t),\dot u (t)]$ has a unique stationary measure $\mu$, and the law of any solution converges to $\mu$ with exponential rate in the dual-Lipschitz norm.