A note on regularity and separation for the stochastic Allen-Cahn equation with logarithmic potential

Abstract

We prove refined space-time regularity for the classical stochastic Allen-Cahn equation with logarithmic potential. This allows to establish a random separation property, i.e. that the trajectories of the solution are strictly separated from the potential barriers. The present contribution extends the results obtained in Bertacco et al. (2022), where separation was proved only for $p$-Laplace operators with $p$ greater than the space-dimension.