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Large deviations for stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term. (English) Zbl 1054.60065

Following a pioneering paper by Freidlin and subsequent work by Sowers as well as Peszat, large deviations upper and lower bounds are established for some stochastic reaction-diffusion systems considered in their small noise limit. The main innovations of the present paper come from the fact that the reaction coefficients are no more assumed to be globally Lipschitz, but only locally Lipschitz and of polynomial growth, whereas no global boundedness assumption is made on the diffusion coefficients, which are rather assumed to be globally Lipschitz. Under such generous hypotheses, several new difficulties have to be overcome in the establishment of a large deviations principle; in particular, the authors are forced to leave the context of Hilbert spaces by working on a space of continuous functions instead of an \(L^2\) space, and providing some exponential estimates in a sup-norm rather than in an \(L^2\) norm. In return, these large deviations results are valid for stochastic reaction-diffusion systems having polynomial reaction terms and an unbounded multiplicative noise term, such as e.g. systems of stochastic Ginzburg-Landau equations with multiplicative noise.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60F10 Large deviations
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