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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
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.