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Mathematics  2015 

Large deviation principle for a stochastic Allen-Cahn equation

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Abstract:

In this paper we consider the Allen-Cahn equation perturbed by a stochastic flux term and prove a large deviation principle. Using an associated stochastic flow of diffeomorphisms the equation can be transformed to a parabolic partial differential equation with random coefficients. We use this structure and first provide a large deviation principle for stochastic flows in function spaces with H\"older-continuity in time. Second, we use a continuity argument and deduce a large deviation principle for the stochastic Allen-Cahn equation.

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