On the density of systems of non-linear spatially homogeneous SPDEs

TL;DR

This paper surveys and extends results on the existence, smoothness, and positivity of densities for systems of non-linear SPDEs driven by Gaussian noise, including applications to heat and wave equations.

Contribution

It extends known results on density properties of single SPDEs to systems, providing conditions for smoothness and positivity of densities in multi-dimensional cases.

Findings

Conditions for existence and smoothness of densities in systems of SPDEs

Criteria for strict positivity of the density in multi-dimensional systems

Application of results to stochastic heat and wave equations

Abstract

In this paper, we consider a system of $k$ second order non-linear stochastic partial differential equations with spatial dimension $d \geq 1$, driven by a $q$-dimensional Gaussian noise, which is white in time and with some spatially homogeneous covariance. The case of a single equation and a one-dimensional noise, has largely been studied in the literature. The first aim of this paper is to give a survey of some of the existing results. We will start with the existence, uniqueness and H\"older's continuity of the solution. For this, the extension of Walsh's stochastic integral to cover some measure-valued integrands will be recalled. We will then recall the results concerning the existence and smoothness of the density, as well as its strict positivity, which are obtained using techniques of Malliavin calculus. The second aim of this paper is to show how these results extend to our system of SPDEs. In particular, we give sufficient conditions in order to have existence and smoothness of the density on the set where the columns of the diffusion matrix span $\R^k$. We then prove that the density is strictly positive in a point if the connected component of the set where the columns of the diffusion matrix span $\R^k$ which contains this point has a non void intersection with the support of the law of the solution. We will finally check how all these results apply to the case of the stochastic heat equation in any space dimension and the stochastic wave equation in dimension $d\in \{1,2,3\}$.