Concentration estimates for slowly time-dependent singular SPDEs on the two-dimensional torus

TL;DR

This paper establishes concentration results for solutions of slowly time-dependent singular SPDEs on the two-dimensional torus, extending finite-dimensional and one-dimensional results to a more complex setting.

Contribution

It extends existing concentration results to two-dimensional torus SPDEs with slow time dependence and discusses bifurcation cases.

Findings

Concentration near stable equilibria in Besov and Hölder norms

Extension of finite-dimensional SDE results to 2D SPDEs

Analysis of bifurcation scenarios in the SPDE context

Abstract

We consider slowly time-dependent singular stochastic partial differential equations on the two-dimensional torus, driven by weak space-time white noise, and renormalised in the Wick sense. Our main results are concentration results on sample paths near stable equilibrium branches of the equation without noise, measured in appropriate Besov and H\"older norms. We also discuss a case involving a pitchfork bifurcation. These results extend to the two-dimensional torus those obtained in [Berglund and Gentz, Proability Theory and Related Fields, 2002] for finite-dimensional SDEs, and in [Berglund and Nader, Stochastics and PDEs, 2022] for SPDEs on the one-dimensional torus.