Large deviation principle for a stochastic Allen–Cahn equation

Martin Heida () and Matthias Röger ()
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Martin Heida: Weierstrass Institute for Applied Analysis and Stochastics
Matthias Röger: Technische Universität Dortmund

Journal of Theoretical Probability, 2018, vol. 31, issue 1, 364-401

Abstract: 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.

Keywords: Large deviations; Stochastic partial differential equations; Stochastic flows; Allen–Cahn equation; 60F10; 60H15; 35R60; 49J45 (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10959-016-0711-7

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