Weak error estimates of Galerkin approximations for the stochastic Burgers equation driven by additive trace-class noise

TL;DR

This paper proves that spectral Galerkin methods for the stochastic Burgers equation achieve weak convergence rates twice as fast as strong rates, supported by new regularity results for related Kolmogorov equations.

Contribution

It provides the first weak convergence rate analysis for spectral Galerkin approximations of the stochastic Burgers equation with additive noise, revealing a rate twice the strong convergence rate.

Findings

Weak convergence rate is approximately 2.

Strong convergence rate is approximately 1.

Regularity results for Kolmogorov equations are established.

Abstract

We establish weak convergence rates for spectral Galerkin approximations of the stochastic viscous Burgers equation driven by additive trace-class noise. Our results complement the known results regarding strong convergence; we obtain essential weak convergence rate 2. As expected, this is twice the known strong rate. The main ingredients of the proof are novel regularity results on the solutions of the associated Kolmogorov equations.