Extended Poisson-Kac Theory: A Unifying Framework for Stochastic Processes with Finite Propagation Velocity

Stochastic processes are a cornerstone of physics and engineering, particularly for modelling transport phenomena in the natural and social sciences, including the random movements of cells, bacteria, and viruses, climate fluctuations or the dynamics of financial markets. Such models typically capture the essential statistical character of dynamics on long timescales, yet many simple models also contain an important flaw, as they fail to ensure finite velocities, thereby violating Einstein’s theory of special relativity.
As LML Fellow Rainer Klages notes in a recent paper written with Massimiliano Giona and Andrea Cairoli, researchers have developed models with finite propagation speeds to address this conceptual problem. So-called Lévy walks (LWs) arise if each jump a random walker takes involves a physically realistic duration of time proportional to the jump length. This requirement leads naturally to finite propagation speeds. Alternatively, so-called Poisson-Kac (PK) processes represent a second fundamental class of dynamics with finite velocities. Such a process is a one-dimensional random walk, wherein the direction of the walker’s velocity reverses at random moments. If these moments follow a Poisson counting process, the intervals between successive changes exhibit an  exponential probability distribution, and the walker’s position obeys a Cattaneo differential equation. In contrast to the classical parabolic diffusion equation, this equation is a hyperbolic diffusion equation stipulating a finite propagation velocity.
Despite their similarities, however, no previous study has examined the cross-links between these two fundamental classes of stochastic processes. In their paper, Klages and colleagues do so, first establishing a clear formal connection between LWs and PK processes. In particular, inspired by the novel formulation of LW dynamics proposed by Fedotov and collaborators, they show how PK processes can be understood as a particular case of LWs. In the process, they develop a comprehensive theory of stochastic processes with finite propagation velocity and finite transition rates. This extended or more general theory offers a flexible approach to modelling many different kinds of dynamical features, as the authors demonstrate in three physically and biologically motivated examples. These stochastic models capture the entire spectrum of diffusive dynamics including normal and anomalous diffusion, the “Brownian yet non-Gaussian” diffusion, as well as more subtle phenomena such as senescence.
The paper is available at https://journals.aps.org/prx/abstract/10.1103/PhysRevX.12.021004

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