Large deviations for invariant measures of general stochastic reaction–diffusion systems

Sandra Cerrai; Michael Röckner

doi:10.1016/j.crma.2003.09.015

Probability Theory

Large deviations for invariant measures of general stochastic reaction–diffusion systems

Presented by: Paul Malliavin

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

Comptes Rendus. Mathématique, Volume 337 (2003) no. 9, pp. 597-602.

Abstract
Résumé

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.

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.

Received: 2003-05-20
Accepted: 2003-09-17
Published online: 2003-10-20

DOI: 10.1016/j.crma.2003.09.015

Author's affiliations:

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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     author = {Sandra Cerrai and Michael R\"ockner},
     title = {Large deviations for invariant measures of general stochastic reaction{\textendash}diffusion systems},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {597--602},
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     year = {2003},
     doi = {10.1016/j.crma.2003.09.015},
     language = {en},
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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/

References
Cited by

[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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