Long-time dynamics of stochastic wave equation with dissipative damping and its full discretization: exponential ergodicity and strong law of large numbers
Meng Cai, Chuchu Chen, Jialin Hong, Tau Zhou
TL;DR
We analyze the stochastic wave equation with dissipative damping $F$ that is not globally Lipschitz in velocity and establish exponential ergodicity for both the continuous dynamics and its full discretization (spectral Galerkin in space plus backward Euler in time). By constructing Lyapunov functionals, we obtain a unique invariant measure and contraction in Wasserstein distance, and we quantify the convergence of numerical invariant measures to the true one in both Wasserstein and weak senses. We also prove strong laws of large numbers for time averages of the exact and numerical solutions, providing rates that connect discretization parameters $(\lambda_N,\tau)$ to ergodic limits. The results yield explicit, quantitative estimates for invariant-measure approximation and justify using long-time time-averaging of a single numerical path to approximate ergodic limits in stochastic wave dynamics.
Abstract
For stochastic wave equation, when the dissipative damping is a non-globally Lipschitz function of the velocity, there are few results on the long-time dynamics, in particular, the exponential ergodicity and strong law of large numbers, for the equation and its numerical discretization to our knowledge. Focus on this issue, the main contributions of this paper are as follows. First, based on constructing novel Lyapunov functionals, we show the unique invariant measure and exponential ergodicity of the underlying equation and its full discretization. Second, the error estimates of invariant measures both in Wasserstein distance and in the weak sense are obtained. Third, the strong laws of large numbers of the equation and the full discretization are obtained, which states that the time averages of the exact and numerical solutions are shown to converge to the ergodic limit almost surely.