Approximation of the invariant law of SPDEs: error analysis using a Poisson equation for a full-discretization scheme

Charles-Edouard Bréhier (1) , Marie Kopec (2)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique
2 IPSO - Invariant Preserving SOlvers

We study the long-time behavior of fully discretized semilinear SPDEs with additive space-time white noise, which admit a unique invariant probability measure $\mu$. 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 $\mu$. 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; 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.

Domaines: Analyse numérique [math.NA]; Probabilités [math.PR]

Charles-Edouard Bréhier, Marie Kopec. Approximation of the invariant law of SPDEs: error analysis using a Poisson equation for a full-discretization scheme. 2013. ⟨hal-00910323v1⟩