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Large deviation principle for a stochastic Allen–Cahn equation

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Abstract

The Allen–Cahn equation is a prototype model for phase separation processes, a fundamental example of a nonlinear spatial dynamic and an important approximation of a geometric evolution equation by a reaction–diffusion equation. Stochastic perturbations, especially in the case of additive noise, to the Allen–Cahn equation have attracted considerable attention. We consider here an alternative random perturbation determined by a Brownian flow of spatial diffeomorphism that was introduced by Röger and Weber (Stoch Partial Differ Equ Anal Comput 1(1):175–203, 2013). We first provide a large deviation principle for stochastic flows in spaces of functions that are Hölder continuous in time, which extends results by Budhiraja et al. (Ann Probab 36(4):1390–1420, 2008). From this result and a continuity argument we deduce a large deviation principle for the Allen–Cahn equation perturbed by a Brownian flow in the limit of small noise. Finally, we present two asymptotic reductions of the large deviation functional.

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Acknowledgments

This work was partially funded by the DFG-Forschergruppe 718 Analysis and Stochastics in Complex Physical Systems. We thank Hendrik Weber for helpful discussions.

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Authors and Affiliations

  1. Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117, Berlin, Germany

    Martin Heida

  2. Fakultät für Mathematik, Technische Universität Dortmund, Vogelpothsweg 87, 44227, Dortmund, Germany

    Matthias Röger

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  1. Martin Heida
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  2. Matthias Röger
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Correspondence to Matthias Röger.

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Heida, M., Röger, M. Large deviation principle for a stochastic Allen–Cahn equation. J Theor Probab 31, 364–401 (2018). https://doi.org/10.1007/s10959-016-0711-7

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  • DOI: https://doi.org/10.1007/s10959-016-0711-7

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