Random separation property for stochastic Allen-Cahn-type equations

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 variational solutions. Then, show that random separation property holds, i.e. almost every trajectory is strictly separated in space time from potential barriers. The threshold random, further provide exponential estimates probability Eventually, exhibit convergence-in-probability result for towards deterministic one, as vanishes, obtain an estimate convergence rate.