Large deviations for white-noise driven, nonlinear stochastic PDEs in two and three dimensions

Martin Hairer; Hendrik Weber

doi:10.5802/afst.1442

Martin Hairer ; Hendrik Weber

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 24 (2015) no. 1, pp. 55-92.

Résumé
Abstract

Nous étudions l’équation d’Allen-Cahn stochastique conduite par un bruit d’intensité $\sqrt{ε}$ et de longueur de corrélation $δ$ en dimensions spatiales deux et trois. Nous considérons la limite $δ, ε \to 0$ et nous décrivons complètement le comportement des grandes déviations associées, suivant les relations entre $δ$ et $ε$ .

La théorie des structures de régularité récemment développée permet d’analyser le comportement des solutions à intensité de bruit $ε$ fixée dans la limite $δ \to 0$ . Un fait crucial est que, afin d’obtenir des limites non-triviales dans cette limite, il est nécessaire d’introduire des contretermes divergents. La théorie des structures de régularité permet d’analyser rigoureusement de telles procédures de renormalisation pour un nombre d’équations intéressantes.

Notre résultat principal est un principe de grandes déviations pour ces équations renormalisées. Il est alors intéressant de noter que les constantes de renormalisation divergentes disparaissent au niveau de la fonction de taux. Une conséquence de ce résultat est une condition optimale sur le comportement relatif de $δ$ et $ε$ qui garantit l’existence d’un principe de grandes déviations également pour l’équation non-renormalisée dans certains régimes.

We study the stochastic Allen-Cahn equation driven by a noise term with intensity $\sqrt{ε}$ and correlation length $δ$ in two and three spatial dimensions. We study diagonal limits $δ, ε \to 0$ and describe fully the large deviation behaviour depending on the relationship between $δ$ and $ε$ .

The recently developed theory of regularity structures allows to fully analyse the behaviour of solutions for vanishing correlation length $δ$ and fixed noise intensity $ε$ . One key fact is that in order to get non-trivial limits as $δ \to 0$ , it is necessary to introduce diverging counterterms. The theory of regularity structures allows to rigorously analyse this renormalisation procedure for a number of interesting equations.

Our main result is a large deviation principle for these renormalised solutions. One interesting feature of this result is that the diverging renormalisation constants disappear at the level of the large deviations rate function. We apply this result to derive a sharp condition on $δ, ε$ that guarantees a large deviation principle for diagonal schemes $ε, δ \to 0$ for the equation without renormalisation.

MR

DOI : 10.5802/afst.1442

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     author = {Martin Hairer and Hendrik Weber},
     title = {Large deviations for white-noise driven, nonlinear stochastic {PDEs} in two and three dimensions},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {55--92},
     publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 24},
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Martin Hairer; Hendrik Weber. Large deviations for white-noise driven, nonlinear stochastic PDEs in two and three dimensions. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 24 (2015) no. 1, pp. 55-92. doi : 10.5802/afst.1442. https://afst.centre-mersenne.org/articles/10.5802/afst.1442/

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