Sharp interface limit for invariant measures of a stochastic Allen-Cahn equation

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

The invariant measure of a one-dimensional Allen-Cahn equation with an additive space-time white noise is studied. This measure is absolutely continuous with respect to a Brownian bridge with a density that can be interpreted as a potential energy term. We consider the sharp interface limit in this setup. In the right scaling this corresponds to a Gibbs-type measure on a growing interval with decreasing temperature. Our main result is that in the limit we still see exponential convergence towards a curve of minimizers of the energy if the interval does not grow too fast. In the original scaling, the measure is concentrated on configurations with precisely one jump.

Original languageEnglish
Pages (from-to)1071-1109
Number of pages39
JournalCommunications on Pure and Applied Mathematics
Volume63
Issue number8
DOIs
Publication statusPublished - 1 Aug 2010

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this