Stochastic Allen–Cahn equation with mobility

We introduce a class of stochastic Allen–Cahn equations with a mobility coefficient and colored noise. For initial data with finite free energy, we analyze the corresponding Cauchy problem on the d-dimensional torus in the time interval [0, T]. Assuming that d ≤ 3 and that the potential has quartic growth, we prove existence and uniqueness of the solution as a process u in L^2 with continuous paths, satisfying almost surely the regularity properties u ∈ C([0, T];H^1) and u ∈ L^2([0, T];H^2)