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

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 with so-called Dobrushin boundary conditions, i.e., inhomogeneous $\pm 1$ Dirichlet boundary conditions that enforce at least one transition layer 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.

Specifically, in the context of system sizes that are exponential with respect to the inverse noise strength---up to the ``critical'' exponential size predicted by the heuristics---we study the extremely strained large deviation event of seeing \emph{more than the one transition layer} between $\pm 1$ that is forced by the boundary conditions. We capture the competition between energy and entropy through upper and lower bounds on the  probability of these unlikely extra transition layers. Our 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. Our second result then studies the distribution of the transition layer. In particular, we establish that, on a super-logarithmic scale, the position of the transition layer is approximately uniformly distributed.

In our arguments we use local large deviation bounds, the strong Markov property, the symmetry of the potential, and measure-preserving reflections.