Comptes Rendus
no. 9
Probability Theory
Large deviations for invariant measures of general stochastic reaction–diffusion systems
[Grandes déviations pour les mesures invariantes de systèmes généraux d'équations de réaction–diffusion stochastiques]

Présenté par : Paul Malliavin

Sandra Cerrai1 ; Michael Röckner2
1 Dip. di Matematica per le Decisioni, Università di Firenze, Via C. Lombroso 6/17, 50134 Firenze, Italy
2 Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany
Comptes Rendus. Mathématique, Volume 337 (2003) no. 9, pp. 597-602.

Dans cet article on prouve un principe de grandes déviations pour les mesures invariantes de systèmes de réaction–diffusion stochastiques dans des domaines bornés de ℝ d ,d⩾1, perturbés par un bruit multiplicatif. On considère des termes de réaction qui ne sont pas Lipschitz-continus et des coefficients de diffusion qui ne sont pas bornés et peuvent être dégénérés.

In this paper we prove a large deviations principle for the invariant measures of a class of reaction–diffusion systems in bounded domains of ℝ d ,d⩾1, perturbed by a noise of multiplicative type. We consider reaction terms which are not Lipschitz-continuous and diffusion coefficients in front of the noise which are not bounded and may be degenerate.

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DOI : 10.1016/j.crma.2003.09.015
Sandra Cerrai 1 ; Michael Röckner 2

1 Dip. di Matematica per le Decisioni, Università di Firenze, Via C. Lombroso 6/17, 50134 Firenze, Italy
2 Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany 
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Sandra Cerrai; Michael Röckner. Large deviations for invariant measures of general stochastic reaction–diffusion systems. Comptes Rendus. Mathématique, Volume 337 (2003) no. 9, pp. 597-602. doi : 10.1016/j.crma.2003.09.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.09.015/

[1] R. Azencott Grandes déviations et applications, École d'Été de Probabilités de Saint-Flour VII, Lecture Notes in Math., 774, Springer-Verlag, 1980

[2] S. Cerrai Second Order PDE's in Finite and Infinite Dimension. A Probabilistic Approach, Lecture Notes in Math., 1762, Springer-Verlag, 2001

[3] S. Cerrai Stochastic reaction–diffusion systems with multiplicative noise and non-Lipschitz reaction term, Probab. Theory Related Fields, Volume 125 (2003), pp. 271-304

[4] S. Cerrai, M. Röckner, Large deviations for stochastic reaction–diffusion systems with multiplicative noise and non-Lipschitz reaction term, Ann. Probab., in press

[5] M.I. Freidlin Random perturbations of reaction–diffusion equations: the quasi deterministic approximation, Trans. Amer. Math. Soc., Volume 305 (1988), pp. 665-697

[6] R. Sowers Large deviations for a reaction–diffusion equation with non-Gaussian perturbation, Ann. Probab., Volume 20 (1992), pp. 504-537

[7] R. Sowers Large deviations for the invariant measure of a reaction–diffusion equation with non-Gaussian perturbations, Probab. Theory Related Fields, Volume 92 (1992), pp. 393-421

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