Malliavin calculus for the stochastic Cahn–Hilliard/Allen–Cahn equation with unbounded noise diffusion

Abstract

The stochastic partial differential equation analyzed in this work, is motivated by a simplified mesoscopic physical model for phase separation. It describes pattern formation due to adsorption and desorption mechanisms involved in surface processes, in the presence of a stochastic driving force. This equation is a combination of Cahn–Hilliard and Allen–Cahn type operators with a multiplicative, white, space–time noise of unbounded diffusion. We apply Malliavin calculus, in order to investigate the existence of a density for the stochastic solution u. In dimension one, according to the regularity result in [5], u admits continuous paths a.s. Using this property, and inspired by a method proposed in [8], we construct a modified approximating sequence for u, which properly treats the new second order Allen–Cahn operator. Under a localization argument, we prove that the Malliavin derivative of u exists locally, and that the law of u is absolutely continuous, establishing thus that a density exists.