Stochastic Representations for the Wave Equation on Graphs and their Scaling Limits

TL;DR

This paper introduces a particle system that probabilistically models the wave equation on graphs, establishes a Feynman-Kac formula, and demonstrates the system's high-density limit converging to the classical wave equation with a phase transition.

Contribution

It provides a novel probabilistic interpretation of the wave equation on graphs and rigorously connects the particle system to the classical wave equation via scaling limits.

Findings

Established a Feynman-Kac-type formula for the system

Proved the high-density limit converges to the wave equation in Euclidean space

Identified a phase transition indicating the sharpness of the scaling limit

Abstract

This paper is devoted to an interacting particle system that provides probabilistic interpretation of the wave equation on graphs. A Feynman-Kac-type formula is established, connecting the expectation of the process with the wave equation on graphs. Non-asymptotic $L^2$ estimates are presented. It is then shown that the high-density hydrodynamic limit of the system is given by the wave equation in Euclidean space. The sharpness of scaling limit result is demonstrated by a phase transition phenomenon.