Absolute continuity of the law of the solution to the stochastic Cahn-Hilliard equation driven by fractional noise

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 spatial dimensions $d\in \{1,2,3\}$ with $d<4H$, where $H\in (\frac{1}{2},1)$ is the Hurst parameter of the noise, and apply Malliavin calculus to investigate the existence of a density for the solution $u$. In particular, we show that $u$ admits a jointly continuous modification 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 $\mathbb{R}$, thereby establishing that a density exists. We obtain the result for the whole range of dimensions and Hurst parameters in which the mild solution is known to exist. The analysis rests on three new estimates. The first bounds the expectation of the $p$-th power ($p\ge 2$) of the $L^{\infty }(D)$-norm of the stochastic integral appearing in the mild formulation. The other two concern the integral involving the operator associated with the kernel that appears in the integral representation of the fractional noise: an upper estimate and, more importantly, a lower estimate on short time intervals, which yields the nondegeneracy of the Malliavin derivative.

arXiv commentThis version extends our previous result also to dimensions 2 and 3