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Title: The sharp interface limit of the stochastic Allen-Cahn equation
Author: Weber, Simon
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 2014
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We study the Allen-Cahn equation ut = ϵ2uxx + f(u) + ϵγW with an additive noise term ϵγW for small ϵ > 0, and in particular the limit ϵ → 0. This is a reaction-diffusion equation, where f (∙) is the negative derivative of a symmetric double-well potential. We study this equation in the interval (0, 1) with symmetric boundary conditions, and relatively general initial conditions, for W we take space-time white noise. Brassesco et al., Funaki and other authors showed (with different boundary conditions) that if we can project the solution of the equation to an energy-optimal deterministic solution with just one zero, then in the sharp interface limit ϵ → 0 of the solution appropriately rescaled in time is a standard Brownian motion. In this work, we extend these results to a much more general case: We start with fairly general initial conditions, show that after some time we are able to project the solution onto energy-optimal deterministic solutions with finitely many zeroes, after which we derive a semimartingale representation for the interfaces; this representation holds until two interfaces get close to each other and annihilate. In the sharp interface limit ϵ → 0, the appropriately time-rescaled interface position of the solution converges weakly to annihilating independent standard Brownian motions. We also derive an analogous result for smooth noise with trace-class covariance operator, in this case the phenomenon happens on a different timescale than for space-time white noise.
Supervisor: Not available Sponsor: Engineering and Physical Sciences Research Council (EPSRC)
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics