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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?

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  • $\begingroup$ For an alternative approach see the follow up work "Intermittency for the wave and heat equations with fractional noise in time" arxiv.org/abs/1311.0021, where they mention the above functional-path formula. $\endgroup$ Commented Jun 25, 2023 at 3:30
  • $\begingroup$ to be clear, it seems unlikely that the wave equation has some Brownian motion representation because in the proof of Feyman-Kac we use the Ito-formula which involves single-time derivative, not double. $\endgroup$ Commented Jun 25, 2023 at 3:32
  • $\begingroup$ As they mention in cs.fsu.edu/~mascagni/MCPDE_FSU.pdf "I In general MCMs for hyperbolic PDEs (like the wave equation: utt = c 2uxx ) are hard to derive as Brownian motion is fundamentally related to diffusion (parabolic PDEs) and to the equilibrium of diffusion processes (elliptic PDEs), in contrast hyperbolic problems model distortion free information propagation which is fundamentally nonrandom" $\endgroup$ Commented Jun 25, 2023 at 3:35
  • $\begingroup$ You need two initial conditions for the wave equation. I understand that $u(0,x)=f(x)$, but what is $u_t(0,x)$ ? $\endgroup$ Commented Sep 8, 2023 at 18:33

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Transferring comments to answer.

For an alternative approach see the follow up work "Intermittency for the wave and heat equations with fractional noise in time" arxiv.org/abs/1311.0021, where they mention the above functional-path formula.

To be clear, it seems unlikely that the wave equation has some Brownian motion representation because in the proof of Feyman-Kac we use the Ito-formula which involves single-time derivative, not double.

As they mention in Monte Carlo Methods for Partial Differential Equations

"In general MCMs for hyperbolic PDEs (like the wave equation: utt = c 2uxx are hard to derive as Brownian motion is fundamentally related to diffusion (parabolic PDEs) and to the equilibrium of diffusion processes (elliptic PDEs), in contrast hyperbolic problems model distortion free information propagation which is fundamentally nonrandom"

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