Malliavin Calculus for the stochastic Cahn-Hilliard equation driven by fractional noise

TL;DR

This paper applies Malliavin calculus to the one-dimensional stochastic Cahn-Hilliard equation driven by fractional noise, proving the existence of a density for its solution and analyzing key stochastic integrals involved.

Contribution

It introduces a novel analysis of the stochastic integral in the mild formulation, establishing the existence of a density for the solution driven by fractional noise.

Findings

Solution paths are continuous almost surely.

The law of the solution admits a density with respect to Lebesgue measure.

Sharp estimates for stochastic integrals are derived.

Abstract

The stochastic partial differential equation analyzed in this work is the Cahn-Hilliard equation perturbed by an additive fractional white noise (fractional in time and white in space). We work in the case of one spatial dimension and apply Malliavin calculus to investigate the existence of a density for the stochastic solution $u$. In particular, we show that $u$ admits continuous paths almost surely and construct a localizing sequence through which we prove that its Malliavin derivative exists locally, and that its law is absolutely continuous with respect to the Lebesgue measure on $\bf R$, establishing thus that a density exists. A key contribution of this work is the analysis of the stochastic integral appearing in the mild formulation: we derive sharp estimates for the expectation of the $p$-th power ($p \geq 2$) of the $L^{\infty}(D)$-norm of this stochastic integral as well as for the integral involving the $L^{\infty}(D)$-norm of the operator associated with the kernel appearing in the integral representation of the fractional noise, all of which are essential for this study.