APPROXIMATION OF THE INVARIANT LAW OF SPDES: ERROR ANALYSIS USING A POISSON EQUATION FOR A FULL-DISCRETIZATION SCHEME

Abstract. We study the long-time behavior of fully discretized semilinear SPDEs with additive space-time white noise, which admit a unique invariant probability measure µ. We show that the average of regular enough test functions with respect to the (possibly non unique) invariant laws of the approximations are close to the corresponding quantity for µ. More precisely, we analyze the rate of the convergence with respect to the different discretization parameters. Here we focus on the discretization in time thanks to a scheme of Euler type, and on a Finite Element discretization in space. The results rely on the use of a Poisson equation, generalizing the approach of [21]; we obtain that the rates of convergence for the invariant laws are given by the weak order of the discretization on finite time intervals: order 1/2 with respect to the time-step and order 1 with respect to the mesh-size. 1