Random separation property for stochastic Allen-Cahn-type equations

TL;DR

This paper investigates stochastic Allen-Cahn equations with singular potentials, establishing existence, regularity, and a random separation property, along with probabilistic estimates and convergence results as noise diminishes.

Contribution

It introduces a novel analysis of the random separation property for stochastic Allen-Cahn equations with singular potentials, including probabilistic bounds and convergence rates.

Findings

Existence and uniqueness of solutions under certain noise conditions

Almost sure spatial and temporal separation from potential barriers

Convergence of the random separation threshold to the deterministic one as noise vanishes

Abstract

We study a large class of stochastic $p$-Laplace Allen-Cahn equations with singular potential. Under suitable assumptions on the (multiplicative-type) noise we first prove existence, uniqueness, and regularity of variational solutions. Then, we show that a random separation property holds, i.e. almost every trajectory is strictly separated in space and time from the potential barriers. The threshold of separation is random, and we further provide exponential estimates on the probability of separation from the barriers. Eventually, we exhibit a convergence-in-probability result for the random separation threshold towards the deterministic one, as the noise vanishes, and we obtain an estimate of the convergence rate.