Smoothness of the joint density for spatially homogeneous SPDEs

TL;DR

This paper investigates the smoothness and positivity of the joint density of solutions to a class of spatially homogeneous second order SPDEs driven by Gaussian noise, using Malliavin calculus techniques.

Contribution

It establishes conditions under which the joint density of the solution is smooth and strictly positive, extending understanding of SPDE regularity properties.

Findings

Density of the solution is smooth under regularity and non-degeneracy conditions.

The density is strictly positive inside the support of the law.

Results apply to a broad class of spatially homogeneous SPDEs.

Abstract

In this paper we consider a general class of second order stochastic partial differential equations on $\mathbb{R}^d$ driven by a Gaussian noise which is white in time and it has a homogeneous spatial covariance. Using the techniques of Malliavin calculus we derive the smoothness of the density of the solution at a fixed number of points $(t,x_1), \dots, (t,x_n)$, $t>0$, assuming some suitable regularity and non degeneracy assumptions. We also prove that the density is strictly positive in the interior of the support of the law.