Stochastic process leading to wave equations in dimensions higher than one

TL;DR

This paper introduces stochastic processes that generate classical wave, telegraph, and Klein-Gordon equations in higher dimensions by incorporating spatial derivative-dependent transitions, extending previous models.

Contribution

It presents a novel stochastic model where velocity transitions depend on spatial derivatives, enabling wave equations in higher dimensions while sacrificing single-particle interpretability.

Findings

Models recover wave equations in arbitrary dimensions

Transitions depend on spatial derivatives of other states

Imitates Huygens' principle in stochastic framework

Abstract

Stochastic processes are proposed whose master equations coincide with classical wave, telegraph, and Klein-Gordon equations. Similar to predecessors based on the Goldstein-Kac telegraph process, the model describes the motion of particles with constant speed and transitions between discreet allowed velocity directions. A new ingredient is that transitions into a given velocity state depend on spatial derivatives of other states populations, rather than on populations themselves. This feature requires the sacrifice of the single-particle character of the model, but allows to imitate the Huygens' principle and to recover wave equations in arbitrary dimensions.