Large deviations for invariant measures of stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term
type: Item
instance of: scholarly article
title: Large deviations for invariant measures of stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term
zbMATH Open document ID: 1066.60029
author: Sandra Cerrai
author: Michael Röckner
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 2005-03-08T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77838
review text: The authors prove a large deviations principle, as \(\varepsilon\) tends to zero, for the family of invariant measures of stochastic reaction-diffusion systems of the form \[ \begin{aligned} \frac{\partial u_i}{\partial t}(t,\xi) = &\mathcal{A}_i u_i(t,\xi)+f_i(\xi,u_1(t,\xi),\dots,u_r(t,\xi))+\\ & + \varepsilon\sum_{j=1}^r g_{ij}(\xi,u_1(t,\xi),\dots,u_r(t,\xi))\mathcal{Q}_j \frac{\partial w_j}{\partial t}(t,\xi),\end{aligned} \] \(t\geq 0\), \(\xi\in \bar{\mathcal{O}}\), with some initial and boundary conditions. \(\mathcal{O}\) is a bounded open set in \(\mathbb{R}^d\), \(d\geq 1\), the differential operators \(\mathcal{A}_i\) are uniformly elliptic, \(\mathcal{Q}_j\) are bounded operators on \(L^2(\mathcal{O})\) and the random perturbations \({\partial w_j}/{\partial t}\) are independent cylindrical Wiener processes. The drift \(f\) is locally Lipschitz and has polynomial growth and \(g\) is globally Lipschitz and may be degenerate. Due to the hypothesis on \(f\), \(g\) and the general space dimension \(d\geq 1\), the authors make use of new techniques and generalise related results of \textit{R. B. Sowers} [Ann. Probab. 20, No.~1, 504--537 (1992; Zbl 0749.60059) and Probab. Theory Relat. Fields 92, No.~3, 393--421 (1992; Zbl 0767.60025)].
reviewed by: Lluís Quer-Sardanyons
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60H15
zbMATH DE Number: 2141606
zbMATH Keywords: large deviations principle
zbMATH Keywords: stochastic partial differential equations
zbMATH Keywords: invariant measures
MaRDI profile type: Publication
OpenAlex ID: W2069441636
DOI: 10.1016/J.ANIHPB.2004.03.001
description: scientific article; zbMATH DE number 2141606