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.
Funding Statement
The third author has been partially funded by MIUR-PRIN Grant 2020F3NCPX “Mathematics for industry 4.0 (Math4I4)”.
Acknowledgments
We are especially grateful to Prof. Ulisse Stefanelli for the insightful discussions about separation properties for deterministic doubly nonlinear evolution equations. The second and third authors are members of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA), Istituto Nazionale di Alta Matematica (INdAM).
Citation
Federico Bertacco. Carlo Orrieri. Luca Scarpa. "Random separation property for stochastic Allen-Cahn-type equations." Electron. J. Probab. 27 1 - 32, 2022. https://doi.org/10.1214/22-EJP830
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