Fractional SPDEs driven by spatially correlated noise: existence of the solution and smoothness of its density

Abstract

In this paper we study a class of stochastic partial differential equations in the whole space ${\mathbb{R}}^d$, with arbitrary dimension $d\ge 1$, driven by a Gaussian noise white in time and correlated in space. The differential operator is a fractional derivative operator. We show the existence, uniqueness and Hölder's regularity of the solution. Then by means of Malliavin calculus, we prove that the law of the solution has a smooth density with respect to the Lebesgue measure.

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• Macroscopic multi-fractality of Gaussian random fields and linear SPDEs with colored noise