We prove that an averaging principle holds for a general class of stochastic reactiondiffusion systems, having unbounded multiplicative noise, in any space dimension. We show that the classical Khasminskii approach for systems with a finite number of degrees of freedom can be extended to infinite dimensional systems. *

We consider the averaging principle for stochastic reaction-diffusion equations. Under some assumptions providing existence of a unique invariant measure of the fast motion with the frozen slow component, we calculate limiting slow motion. The study of solvability of Kolmogorov equations in Hilbert spaces and the analysis of regularity properties of solutions, allow to generalize the classical approach to finite-dimensional problems of this type in the case of SPDE's.

We study existence and uniqueness of a mild solution in the space of continuous functions and existence of an invariant measure for a class of reaction-diffusion systems on bounded domains of R d , perturbed by a multiplicative noise. The reaction term is assumed to have polynomial growth and to be locally Lipschitz-continuous and monotone. The noise is white in space and time if d = 1 and coloured in space if d > 1; in any case the covariance operator is never assumed to be Hilbert-Schmidt. The multiplication term in front of the noise is assumed to be Lipschitz-continuous and no restrictions are given either on its linear growth or on its degenaracy. Our results apply, in particular, to systems of stochastic Ginzburg-Landau equations with multiplicative noise.

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 Lipschitzcontinuous and diffusion coefficients in front of the noise which are not bounded and may be degenerate. This covers for example the case of Ginzburg-Landau systems with unbounded and possibly degenerate multiplicative noise.  2004 Elsevier SAS. All rights reserved. RésuméDans 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 lipschitziens et des coefficients de diffusion qui ne sont pas bornés et peuvent être dégénérés. Ceci s'applique par exemple au cas de systèmes de Ginzburg-Landau avec bruit multiplicatif non borné et éventuellement dégénéré.  2004 Elsevier SAS. All rights reserved. MSC: 60F10; 60H15

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