Characterization of the second order random fields subject to linear distributional PDE constraints

Let $L$ be a linear differential operator acting on functions defined over an open set $\mathcal{D} \subset \mathbb{R}^d$. In this article, we characterize the measurable second order random fields $U = (U(x))_{x \in \mathcal{D}}$ whose sample paths $U_{\omega}$ all verify the partial differential equation (PDE) $L(u) = 0$, solely in terms of their first two moments. When compared to previous similar results, the novelty lies in that the equality $L(u) = 0$ is understood in the \textit{sense of distributions}, which is a powerful functional analysis framework mostly designed to study linear PDEs. This framework enables to reduce to the minimum the required differentiability assumptions over the first two moments of $U$ as well as over its sample paths in order to make sense of the PDE $L(U_{\omega})=0$. In view of Gaussian process regression (GPR) applications, we show that when $U$ is a Gaussian process (GP), the sample paths of $U$ conditioned on pointwise observations still verify the constraint $L(u)=0$ in the distributional sense. We finish by deriving a simple but instructive example, a GP model for the 3D linear wave equation, for which our theorem is applicable and where the previous results from the literature do not apply in general