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Large deviation principle for a stochastic Allen–Cahn equation

Author

Listed:
  • Martin Heida

    (Weierstrass Institute for Applied Analysis and Stochastics)

  • Matthias Röger

    (Technische Universität Dortmund)

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.

Suggested Citation

  • Martin Heida & Matthias Röger, 2018. "Large deviation principle for a stochastic Allen–Cahn equation," Journal of Theoretical Probability, Springer, vol. 31(1), pages 364-401, March.
    • Handle: RePEc:spr:jotpro:v:31:y:2018:i:1:d:10.1007_s10959-016-0711-7
    DOI: 10.1007/s10959-016-0711-7
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    Cited by:

    1. Aguirre, Natham & Kowalczyk, Michał, 2022. "Large deviations approach to a one-dimensional, time-periodic stochastic model of pattern formation," Chaos, Solitons & Fractals, Elsevier, vol. 165(P1).

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