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 is an element of C([0, T]; H-1) and u is an element of L-2 ([0, T]; H-2).