The Feynman–Kac formula provides a functional (Wiener) integral representation of the solution $u$ to the heat equation \begin{align*} \partial_t u &= \frac{1}{2}\Delta_x u,\\ u(0,x) &= f(x), \end{align*} namely \begin{align*} u(t,x) &= \mathbb{E}[f(B_t+x)]\\ &= \int_{\mathcal{C}([0,t],\mathbb{R})}f(\gamma(t)+x)\,\mathrm{d}W(\gamma). \end{align*} There are a number of similar formulas for other linear parabolic partial differential equations, as well as elliptic PDEs. Meanwhile, Feynman–Kac representations of hyperbolic PDEs seem harder to come by.

**Question.** Is it possible to express the solution to the 1D wave equation as an integral in a space of functions with respect to an appropriate measure?

More generally, what is the situation for other hyperbolic PDEs?

Concerning the first question, I found the following two articles:

Zhang–Yu–Mascagni,

Revisiting Kac’s method: A Monte Carlo algorithm for solving the Telegrapher’s equations.Dalang–Mueller–Tribe,

A Feynman-Kac-type formula for the deterministic and stochastic wave equations, arXiv:0710.2861.

The first article recalls Kac's solution to the 1D telegrapher's equation, which gives also a probabilistic solution to the 1d wave equation \begin{align*} \partial^2_t u &= c^2\Delta_x u,\\ u(0,x) &= f(x), \end{align*} namely $$u(t,x)=\frac{1}{2}\left(\mathbb{E}\left[f\left(x+\int^{t}_{0}c(-1)^{N(\tau)}\,\mathrm{d}\tau\right)\right]+\mathbb{E}\left[f\left(x-\int^{t}_{0}c(-1)^{N(\tau)}\,\mathrm{d}\tau\right)\right]\right),$$ where $N(t)$ is a Poisson process.

Meanwhile, Equation (3.1) in the second article gives another probabilistic representation of the solution to the 1D wave equation (along with other equations, such as the beam equation), also involving Poisson processes.

However, is it possible to rewrite these two probabilistic representations as functional integrals, as in the case of the Feynman–Kac formula for the heat equation?