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

TL;DR

This paper applies Malliavin calculus to a stochastic PDE combining Cahn-Hilliard and Allen-Cahn operators with unbounded noise, proving the existence of a probability density for its solution in one dimension.

Contribution

It introduces a novel approach to establish the existence of a density for solutions of a complex stochastic PDE with unbounded noise diffusion, combining regularity and localization techniques.

Findings

Proves the solution's law is absolutely continuous in one dimension.

Constructs a modified approximation sequence for the solution.

Establishes local Malliavin differentiability of the solution.

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 \cite{AKM}, $u$ admits continuous paths a.s. Using this property, and inspired by a method proposed in \cite{CW1}, 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.