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

TL;DR

This paper investigates fractional stochastic PDEs with spatially correlated Gaussian noise, establishing existence, uniqueness, regularity, and smoothness of the solution's probability density using Malliavin calculus.

Contribution

It introduces a framework for fractional SPDEs driven by spatially correlated noise and proves the smoothness of the solution's law, extending previous results to more general operators.

Findings

Existence and uniqueness of solutions for the fractional SPDEs.

Hölder regularity of the solutions.

Smoothness of the solution's probability density.

Abstract

In this paper we study a class of stochastic partial differential equations in the whole space $\mathbb{R}^{d}$, with arbitrary dimension $d\geq 1$, 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 H\"{o}lder'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.