  Let $P$ be a linear differential operator over $\mathcal{D} \subset
\mathbb{R}^d$ and $U = (U_x)_{x \in \mathcal{D}}$ a second order stochastic
process. In the first part of this article, we prove a new necessary and
sufficient condition for all the trajectories of $U$ to verify the partial
differential equation (PDE) $T(U) = 0$. This condition is formulated in terms
of the covariance kernel of $U$. When compared to previous similar results, the
novelty lies in that the equality $T(U) = 0$ is understood in the \textit{sense
of distributions}, which is a relevant framework for PDEs. This theorem
provides precious insights during the second part of this article, devoted to
performing "physically informed" machine learning for the homogeneous 3
dimensional free space wave equation. We perform Gaussian process regression
(GPR) on pointwise observations of a solution of this PDE. To do so, we
propagate Gaussian processes (GP) priors over its initial conditions through
the wave equation. We obtain explicit formulas for the covariance kernel of the
propagated GP, which can then be used for GPR. We then explore the particular
cases of radial symmetry and point source. For the former, we derive
convolution-free GPR formulas; for the latter, we show a direct link between
GPR and the classical triangulation method for point source localization used
in GPS systems. Additionally, this Bayesian framework provides a new answer for
the ill-posed inverse problem of reconstructing initial conditions for the wave
equation with a limited number of sensors, and simultaneously enables the
inference of physical parameters from these data. Finally, we illustrate this
physically informed GPR on a number of practical examples.
