Exponential Mixing and Ergodic Theorems for a Damped Nonlinear Wave Equation with Space-Time Localised Noise

摘要

This paper is devoted to study a damped nonlinear wave equation driven by a space-time localised noise, in a bounded domain with a smooth boundary. The equation is supplemented with the Dirichlet boundary conditions. It is assumed that the random perturbation is non-degenerate. We prove that the Markov process generated by the solution possesses a unique stationary distribution which is exponentially mixing. A strong law of large numbers and the central limit theorem are derived for this Markov process and used to estimate the corresponding rates of convergence.