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
[Grandes déviations pour les mesures invariantes de systèmes généraux d'équations de réaction–diffusion stochastiques]

Présenté par : 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.

Résumé
Abstract

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.

Reçu le : 2003-05-20
Accepté le : 2003-09-17
Publié le : 2003-10-20

DOI : 10.1016/j.crma.2003.09.015

Affiliations des auteurs :

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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     pages = {597--602},
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     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/

Bibliographie
Cité par

[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

Christian Kuehn Stochastic Systems, Multiple Time Scale Dynamics, Volume 191 (2015), p. 477 | DOI:10.1007/978-3-319-12316-5_15
Christian Kuehn; Martin G. Riedler Large deviations for nonlocal stochastic neural fields, The Journal of Mathematical Neuroscience, Volume 4 (2014), p. 33 (Id/No 1) | DOI:10.1186/2190-8567-4-1 | Zbl:1291.92038
Sandra Cerrai Stabilization by noise for a class of stochastic reaction-diffusion equations, Probability Theory and Related Fields, Volume 133 (2005) no. 2, pp. 190-214 | DOI:10.1007/s00440-004-0421-4 | Zbl:1077.60046

Cité par 3 documents. Sources : Crossref, zbMATH

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