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.

翻译：对于随机波动方程，当耗散阻尼是速度的非全局Lipschitz函数时，据我们所知，关于该方程及其数值离散的长时动力学，特别是指数遍历性与强大数定律的研究结果较少。针对这一问题，本文的主要贡献如下：首先，通过构造新型Lyapunov泛函，证明了底层方程及其全离散化具有唯一不变测度和指数遍历性；其次，获得了不变测度在Wasserstein距离和弱意义下的误差估计；第三，得到了方程及其全离散化的强大数定律，表明精确解和数值解的时间平均值几乎必然收敛于遍历极限。