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, furt...

In this paper we show that ωBand ωS-regular languages satisfy the following separation-type theorem If L1, L2 are disjoint languages of ω-words both recognised by ωB(resp. ωS)automata then there exists an ω-regular language Lsep that contains L1, and whose complement contains L2. In particular, if a language and its complement are recognised by ωB(resp. ωS)automata then the language is ω-regula...

We say that a space X has the separation property provided that if A and B are subsets of X with A countable and B first category, then there is a homeomorphism f : X → X such that f(A) ∩ B = ∅. We prove that a Borel space with this property is Polish. Our main result is that if the homeomorphisms needed in the separation property for the space X come from the homeomorphisms given by an action ...