Weak backward error analysis for Langevin process

TL;DR

This paper develops a weak backward error analysis for implicit numerical schemes approximating Langevin equations, showing they preserve key statistical properties and exhibit exponential mixing.

Contribution

It introduces a high-order weak backward error analysis for implicit Langevin schemes, linking numerical solutions to modified Kolmogorov equations and invariant measures.

Findings

Numerical generator matches a modified Kolmogorov equation up to high order.

Numerical measures are close to a modified invariant measure.

Implicit scheme exhibits exponential mixing behavior.

Abstract

We consider numerical approximations of stochastic Langevin equations by implicit methods. We show a weak backward error analysis result in the sense that the generator associated with the numerical solution coincides with the solution of a modified Kolmogorov equation up to high order terms with respect to the stepsize. This implies that every measure of the numerical scheme is close to a modified invariant measure obtained by asymptotic expansion. Moreover, we prove that, up to negligible terms, the dynamic associated with the implicit scheme considered is exponentially mixing.