Ergodic semi-implicit approximations to periodic measures of stochastic differential equations with locally Lipschitz drifts—Error analysis in Wasserstein distance

We study numerical approximations to the periodic measures of time-periodic stochastic differential equations. For those systems with locally Lipschitz coefficients, while the explicit Euler-Maruyama scheme does not work, we carry out semi-implicit Euler-Maruyama schemes to compute their periodic measures. We prove the local Doeblin condition for the numerical schemes uniformly with respect to discretization step size. This, together with a Lyapunov function argument due to the weakly dissipative condition, leads to the existence and uniqueness of periodic measures of numerical schemes, and geometric ergodicity with the convergence being independent of the step size in the discretization. The novelty of our approach is that without knowing any a priori information about periodic measure of the original problem, even the existence, we can prove its existence and ergodicity from that of periodic measure of the discretized numerical scheme.