to cross from point $x'$ to point $x$ on the $(d-1)$ dimensional SOS. The phase $S(x,x')$ is the classical action at a fixed frequency $\omega$ of the ray and $\mu$ is a phase index arising due to boundary conditions or ray caustics; $S$ can be complex if there is damping. Note that, when considering the Helmholtz case in a domain $\Omega$, there is only one such ray trajectory from $x'
 \to x$. The wave number $k$ is introduced here for convenience; in the special case of the Helmholtz equation (\[helm\]), we have $S(x,x') \equiv k
\, L(x,x')$, the optical length of the chord connecting $x'$ to $x$.

The correlation function $\hat{\Gamma}$ {#sec:corr}
---------------------------------------

We are primarily interested in classical wave problems driven by noisy, stochastic sources. In this context, the natural solution of the problem is in terms of a two-point correlation function $$\Gamma(x_1,x_2) = \bra{x_1}\hat{\Gamma}\ket{x_2},$$ where $\hat{\Gamma}$ can be interpreted as a density matrix of the system and $x_1$ and $x_2$ are coordinates on the SOS. In the simplest case of coherent driving, we may consider $\hat{\Gamma} = \ket{\psi^-}\bra{\psi^-}$, with $\ket{\psi^-}$ a solution of the form (\[psim\]) for a specific source term and at a given frequency. More generally, 