Relativistic diffusion processes and random walk models

TL;DR

This paper introduces a new relativistic diffusion process that is continuous, prevents superluminal speeds, and aligns with nonrelativistic limits, offering an alternative to existing models like the telegraph equation.

Contribution

A generalized Wiener process is proposed that is consistent with relativity, continuous, non-superluminal, and reduces to the standard Wiener process in the nonrelativistic limit.

Findings

The process is non-Markovian, consistent with relativistic constraints.

The relativistic propagator is derived from the nonrelativistic Wiener propagator.

Provides a viable alternative to the telegraph equation for modeling relativistic diffusion.

Abstract

The nonrelativistic standard model for a continuous, one-parameter diffusion process in position space is the Wiener process. As well-known, the Gaussian transition probability density function (PDF) of this process is in conflict with special relativity, as it permits particles to propagate faster than the speed of light. A frequently considered alternative is provided by the telegraph equation, whose solutions avoid superluminal propagation speeds but suffer from singular (non-continuous) diffusion fronts on the light cone, which are unlikely to exist for massive particles. It is therefore advisable to explore other alternatives as well. In this paper, a generalized Wiener process is proposed that is continuous, avoids superluminal propagation, and reduces to the standard Wiener process in the non-relativistic limit. The corresponding relativistic diffusion propagator is obtained directly from the nonrelativistic Wiener propagator, by rewriting the latter in terms of an integral over actions. The resulting relativistic process is non-Markovian, in accordance with the known fact that nontrivial continuous, relativistic Markov processes in position space cannot exist. Hence, the proposed process defines a consistent relativistic diffusion model for massive particles and provides a viable alternative to the solutions of the telegraph equation.