Weber, Hendrik: Sharp interface limit for the stochastic Allen-Cahn equation. - Bonn, 2010. - Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn.
Online-Ausgabe in bonndoc: https://nbn-resolving.org/urn:nbn:de:hbz:5N-20671
@phdthesis{handle:20.500.11811/4545,
urn: https://nbn-resolving.org/urn:nbn:de:hbz:5N-20671,
author = {{Hendrik Weber}},
title = {Sharp interface limit for the stochastic Allen-Cahn equation},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2010,
month = mar,

note = {The behavior of the Allen-Cahn equation
t u ε (x,t)= Δ u ε (x,t) - ε -2 F'(u ε (x,t))+ ξ t ε
with an additional stochastic term ξ t ε is studied for small values of ε. This equation is a reaction diffusion equation with a particular shape of the reaction term -F' which is the negative derivative of a double-well potential with two wells of equal depth.
In the first part the invariant measure for this equation is studied in the case, where x ∈ [-1,1] takes values in a compact one-dimensional domain, and where ξ t ε denotes a space-time white noise. This measure is absolutely continuous with respect to a Brownian bridge with appropriate boundary conditions. A scaled version of this measure is transformed to a Gibbs-type measure on a growing interval. Then it is shown that these transformed measures concentrate around a one-dimensional curve of minimizers in the infinite-dimensional space of possible configurations. This implies that in the original scaling the measures concentrate on the set of step functions with precisely one jump.
In the second part the dynamical system is studied in higher dimensions. Here the noise term ξ t ε is constant in space, and smoothened in time, with a correlation length that goes to zero at a precise rate as ε↓ 0. In the limit one obtains almost surely configurations that are concentrated on { ± 1 }. The development of the phase boundaries is driven by its mean curvature with an additional stochastic forcing term.},

url = {https://hdl.handle.net/20.500.11811/4545}
}

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