Ergodicity Of The Stochastic Nonlinear Wave Equations

In this paper, we consider the ergodicity of the following three dimensional nonlinear wave equation with additive noise,First of all, we use the classical analysis to prove that the existence and uniqueness of this equation. Then, we consider some priori estimates for the linear problem and nonlin-ear problem of the stochastic wave equations. Therefore, we will obtain the irreducibility property of the Markov semigroup, which associated with the solutions of the stochastic nonlinear wave equation. Meanwhile, we need to prove that it has the asymptotic strong Feller property and the tight property by the Malliavin calculus. In order to prove the asymptotic strong Feller property here, we first assume that we can obtain an estimate under the square sense. Last, we will show that the Markov operator associated with the flow ξu(t)=[u(t),u(t)] has a unique invariant measure by the ergodic theory for the stochastic nonlinear wave equation.