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

Nils Berglund (1, 2) , Rita Nader (1)

1 IDP - Institut Denis Poisson
2 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans

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, PTRF 2002] for finite-dimensional SDEs, and in [Berglund and Nader, Stochastics and Partial Differential Equations, 2022] for SPDEs on the one-dimensional torus.

Mots clés: Stochastic PDEs; sample-path estimates; slow-fast systems; renormalisation.

Domaines: Probabilités [math.PR]; Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]

Nils Berglund, Rita Nader. Concentration estimates for slowly time-dependent singular SPDEs on the two-dimensional torus. Electronic Journal of Probability, 2024, 29 (none), ⟨10.1214/24-EJP1094⟩. ⟨hal-03794269⟩