  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.
