Relativistic Weierstrass random walks

TL;DR

This paper introduces a relativistic extension of the Weierstrass random walk, revealing a crossover from superdiffusive Lévy flights to Gaussian diffusion due to relativistic constraints, with implications for spacetime phenomena.

Contribution

It presents a simple relativistic Markov chain model that demonstrates a transition between superdiffusive and diffusive regimes, highlighting the impact of relativity on stochastic processes.

Findings

Existence of a transition time $t_c$ separating two diffusion regimes

Superdiffusive Lévy flights occur for $t < t_c$

Gaussian diffusion dominates for $t > t_c$

Abstract

The Weierstrass random walk is a paradigmatic Markov chain giving rise to a L\'evy-type superdiffusive behavior. It is well known that Special Relativity prevents the arbitrarily high velocities necessary to establish a superdiffusive behavior in any process occurring in Minkowski spacetime, implying, in particular, that any relativistic Markov chain describing spacetime phenomena must be essentially Gaussian. Here, we introduce a simple relativistic extension of the Weierstrass random walk and show that there must exist a transition time $t_c$ delimiting two qualitative distinct dynamical regimes: the (non-relativistic) superdiffusive L\'evy flights, for $ t < t_c$, and the usual (relativistic) Gaussian diffusion, for $t>t_c$. Implications of this crossover between different diffusion regimes are discussed for some explicit examples. The study of such an explicit and simple Markov chain can shed some light on several results obtained in much more involved contexts.