Numerical Approximation of Stationary Distribution for SPDEs

TL;DR

This paper demonstrates that exponential integrator schemes for certain SPDEs have unique stationary distributions when step sizes are small, and their weak limits match the SPDE's distribution.

Contribution

It establishes the existence and uniqueness of stationary distributions for exponential integrator schemes applied to SPDEs and shows their weak convergence to the SPDE's distribution.

Findings

Existence of unique stationary distribution for the scheme

Weak limit of the scheme's law matches the SPDE's distribution

Stationary distribution exists for sufficiently small stepsize

Abstract

In this paper, we show that the exponential integrator scheme both in spatial discretization and time discretization for a class of stochastic partial differential equations has a unique stationary distribution whenever the stepsize is sufficiently small, and reveal that the weak limit of the law for the exponential integrator scheme is in fact the counterpart for the stochastic partial differential equation considered.