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

TL;DR

The paper addresses weak convergence rates for spectral Galerkin approximations of the 1D stochastic viscous Burgers equation with additive trace-class noise. By deriving new regularity results for the Kolmogorov equations associated with the SPDE and establishing robust exponential moment and space-time regularity estimates, the authors prove a weak convergence rate essentially equal to $2$, twice the known strong rate. The approach hinges on uniform-in-$M$ derivative bounds for the Kolmogorov solutions and a detailed decomposition of the weak error, including random initial data considerations. The findings provide a rigorous benchmark for weak error behavior of Galerkin discretizations in nonlinear SPDEs and lay groundwork for future extensions to fully discrete schemes and other nonlinear parabolic SPDEs.

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.

Theorems & Definitions (37)

• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof