We study a class of stochastic semilinear damped wave equations driven by additive Wiener noise. Owing to the damping term, under appropriate conditions on the nonlinearity, the solution admits a unique invariant distribution. We apply semi-discrete and fully-discrete methods in order to approximate this invariant distribution, using a spectral Galerkin method and an exponential Euler integrator for spatial and temporal discretization respectively. We prove that the considered numerical schemes also admit unique invariant distributions, and we prove error estimates between the approximate and exact invariant distributions, with identification of the orders of convergence. To the best of our knowledge this is the first result in the literature concerning numerical approximation of invariant distributions for stochastic damped wave equations.

翻译：本文研究由加法维纳噪声驱动的一类随机半线性阻尼波动方程。由于阻尼项的存在，在适当的非线性条件下，解具有唯一的不变分布。我们采用半离散和全离散方法逼近该不变分布，分别利用谱伽辽金方法和指数欧拉积分器进行空间和时间离散化。我们证明所考虑的数值格式也具有唯一的不变分布，并建立了近似不变分布与精确不变分布之间的误差估计，同时确定了收敛阶。据我们所知，这是文献中关于随机阻尼波动方程不变分布数值逼近的首个结果。