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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
1/2013

Invariant measure of the stochastic Allen-Cahn equation: the regime of small noise and large system size

Felix Otto, Hendrik Weber and Maria G. Westdickenberg

Abstract

We study the invariant measure of the one-dimensional stochastic Allen-Cahn equation for a small noise strength and a large but finite system. We endow the system with inhomogeneous Dirichlet boundary conditions that enforce at least one transition from $-1$ to $1$. (Our methods can be applied to other boundary conditions as well.) We are interested in the competition between the "energy" that should be minimized due to the small noise strength and the "entropy" that is induced by the large system size.

Our methods handle system sizes that are exponential with respect to the inverse noise strength, up to the "critical" exponential size predicted by the heuristics. We capture the competition between energy and entropy through upper and lower bounds on the probability of extra transitions between $\pm 1$. These bounds are sharp on the exponential scale and imply in particular that the probability of having one and only one transition from $-1$ to $+1$ is exponentially close to one. In addition, we show that the position of the transition layer is uniformly distributed over the system on scales larger than the logarithm of the inverse noise strength.

Our arguments rely on local large deviation bounds, the strong Markov property, the symmetry of the potential, and measure-preserving reflections.

Received:
Jan 3, 2013
Published:
Jan 4, 2013

Related publications

inJournal
2014 Journal Open Access
Felix Otto, Hendrik Weber and Maria G. Westdickenberg

Invariant measure of the stochastic Allen-Cahn equation : the regime of small noise and large system size

In: Electronic journal of probability, 19 (2014), p. 23