Ergodic Numerical Approximation to Periodic Measures of Stochastic Differential Equations

In this paper, we consider numerical approximation to periodic measure of a time periodic stochastic differential equations (SDEs) under weakly dissipative condition. For this we first study the existence of the periodic measure $\rho_t$ and the large time behaviour of $\mathcal{U}(t+s,s,x) := \mathbb{E}\phi(X_{t}^{s,x})-\int\phi d\rho_t,$ where $X_t^{s,x}$ is the solution of the SDEs and $\phi$ is a test function being smooth and of polynomial growth at infinity. We prove $\mathcal{U}$ and all its spatial derivatives decay to 0 with exponential rate on time $t$ in the sense of average on initial time $s$. We also prove the existence and the geometric ergodicity of the periodic measure of the discretized semi-flow from the Euler-Maruyama scheme and moment estimate of any order when the time step is sufficiently small (uniform for all orders). We thereafter obtain that the weak error for the numerical scheme of infinite horizon is of the order $1$ in terms of the time step. We prove that the choice of step size can be uniform for all test functions $\phi$. Subsequently we are able to estimate the average periodic measure with ergodic numerical schemes.