Approximation of the invariant measure with an Euler scheme for Stochastic PDE's driven by Space-Time White Noise

TL;DR

This paper demonstrates that a semi-implicit Euler scheme can effectively approximate the invariant measure of a parabolic stochastic PDE driven by space-time white noise, achieving a convergence order of 1/2.

Contribution

It provides a rigorous analysis of the approximation of invariant measures for stochastic PDEs using a numerical Euler scheme, including convergence rates.

Findings

The scheme converges with order 1/2 in time step.

The stochastic PDE admits a unique ergodic invariant measure.

Numerical approximation preserves ergodic properties.

Abstract

In this article, we consider a stochastic PDE of parabolic type, driven by a space-time white-noise, and its numerical discretization in time with a semi-implicit Euler scheme. When the nonlinearity is assumed to be bounded, then a dissipativity assumption is satisfied, which ensures that the SDPE admits a unique invariant probability measure, which is ergodic and strongly mixing - with exponential convergence to equilibrium. Considering test functions of class $\mathcal{C}^2$, bounded and with bounded derivatives, we prove that we can approximate this invariant measure using the numerical scheme, with order 1/2 with respect to the time step.