Is there a Gaussian process for the solutions of the wave equation?

Question

Call a Gaussian process $g$ a prior for a topological space $X$ if the realizations of $g$ are (a.s.) contained in $X$ and dense.

Consider the 1D wave equation $\frac{\partial^2}{\partial t^2}u(t,x)=\frac{\partial^2}{\partial x^2}u(t,x)$. There exists a Gaussian process prior for the smooth solutions of this equation. To construct this Gaussian process, recall that the smooth solutions of the 1D wave equation are exactly the functions of the form $$u(t,x)=f(t-x)+g(t+x)$$ for $f$, $g$ smooth functions of one parameter. Now, it suffices to assign any Gaussian process prior for the set of smooth functions to $f$ and $g$, e.g., Gaussian processes with zero mean function and squared exponential covariance function. Adding these two Gaussian processes leads to the desired Gaussian process, e.g., zero mean and covariance $$((t_1,x_1),(t_2,x_2)) \mapsto \exp(-\frac12(t_1-x_1-t_2+x_2)^2)+\exp(-\frac12(t_1+x_1-t_2-x_2)^2).$$

Is there a Gaussian process prior for the (smooth) solutions of the higher dimensional wave equations, in particular the 3D wave equation?

I would strongly prefer a covariance function in closed form, and no abstract construction. I am flexible with the corresponding topology, e.g., the usual Frechet topology for the set of smooth functions.

My motivation is the following. Recently, I constructed a Gaussian process prior for the solutions of the inhomogeneous Maxwell's equations (https://arxiv.org/abs/1801.09197, Example 4.1). The construction used the parametrization of this solution set via the 4 potentials, where the 4 potentials are arbitrary functions. I wondered whether there is also a Gaussian process prior for the set of solutions of the homogeneous Maxwell's equations. Such a prior exists under a positive answer to my question, as the homogeneous Maxwell's equations are parametrized by solutions of the 3D wave equation.

Answer 1

Sounds like what you are after is a way to represent the solutions of the wave equation in terms of some "free data", meaning some number of functions that are not constrained by any further equations. Well, this is precisely what the initial value problem accomplishes. Any solution $u(t,x,y)$ of the 3D wave equation can be uniquely determined by its initial data $u(0,x,y) = f(x,y)$ and $\partial_t u(0,x,y) = g(x,y)$. If $u(t,x,y)$ is smooth, then so are $f(x,y)$ and $g(x,y)$. Conversely, for any smooth $f(x,y)$ and $g(x,y)$ there is a corresponding smooth solution $u(t,x,y)$. So your free data can taken as the functions $f(x,y)$ and $g(x,y)$, for which you should have no problem constructing Gaussian priors.

The explicit relationship between $u(t,x,y)$ is through an integral formula: $$u(t,x,y) = \int_{\mathbb{R}^2} dx' dy' [F(t,x,y;x',y') f(x',y') + G(t,x,y;x',x') g(x',y')]$$ where $F$ and $G$ are certain distributions. The precise form of $F$ and $G$ is given by Kirchhoff's formulae, known in any dimension. Perhaps this representation is explicit enough for your purposes.