Random separation property for stochastic AllenCahntype equations
Abstract
We study a large class of stochastic $p$Laplace AllenCahn equations with singular potential. Under suitable assumptions on the (multiplicativetype) 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 convergenceinprobability result for the random separation threshold towards the deterministic one, as the noise vanishes, and we obtain an estimate of the convergence rate.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.06544
 Bibcode:
 2021arXiv211006544B
 Keywords:

 Mathematics  Probability;
 Mathematics  Analysis of PDEs