Large deviations for white-noise driven, nonlinear stochastic PDEs in two and three dimensions
type: Item
instance of: scholarly article
title: Large deviations for white-noise driven, nonlinear stochastic PDEs in two and three dimensions
zbMATH Open document ID: 1319.60141
author: Hendrik Weber
author: Martin Hairer
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 2015-03-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/1404.5863
review text: The authors consider the stochastic Allen-Cahn equation \[ \partial_t u=\Delta u+Cu-u^3+\sqrt{\varepsilon}\;\xi_\delta, \] where \(C\in\mathbb{R}\) and \(\xi_\delta\) is a random driving noise with weak intensity \(\sqrt{\varepsilon}\) and almost white, i.e., the correlation length scale is \(\delta\ll 1\). The limiting behaviour of the solutions is studied as \(\varepsilon\) and \(\delta\) approach zero. The theory of regularity structures is used to analyse the behaviour of the solutions for fixed \(\varepsilon\) and \(\delta\rightarrow 0\). In order to obtain nontrivial limits as \(\delta\) vanishes, they consider the modified equation \[ \partial_t u=\Delta u+(C+3\varepsilon C_\delta^{(1)}-9\varepsilon^2 C_\delta^{(2)})u-u^3+\sqrt{\varepsilon}\;\xi_\delta. \] The main result of the paper states a large deviation principle for these renormalised solutions. A remarkable feature is that the diverging renormalisation constants \(C_\delta^{(1)}\) and \(C_\delta^{(2)}\) disappear on the level of large deviations. This result is applied to derive a sharp condition on \(\delta,\varepsilon\) that guarantees a large deviation principle for the equation without renormalisation.
reviewed by: Dora Seleši
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 81T17
zbMATH DE Number: 6418003
zbMATH Keywords: stochastic partial differential equations
zbMATH Keywords: large deviations
zbMATH Keywords: white noise
zbMATH Keywords: regularity structures
Wikidata QID: Q56894161
MaRDI profile type: Publication
OpenAlex ID: W1496994543
arXiv ID: 1404.5863
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DOI: 10.5802/AFST.1442
description: scientific article