Stochastic foundations of undulatory transport phenomena: Generalized Poisson-Kac processes - Part I Basic theory

TL;DR

This paper introduces Generalized Poisson-Kac processes, extending telegrapher's noise with Markovian velocity switching, highlighting their regular trajectories and convergence to Brownian motion, with physical implications discussed in subsequent parts.

Contribution

It defines the structural properties of GPK processes, generalizing Poisson-Kac processes with Markovian velocity switching, and explores their fundamental mathematical features.

Findings

GPK processes have trajectory regularity almost everywhere.

GPK processes converge to Brownian motion in the Kac limit.

GPK processes generalize telegrapher's noise with Markovian velocity switching.

Abstract

This article introduces the notion of Generalized Poisson-Kac (GPK) processes which generalize the class of "telegrapher's noise dynamics" introduced by Marc Kac in 1974, usingPoissonian stochastic perturbations. In GPK processes the stochastic perturbation acts as a switching amongst a set of stochastic velocity vectors controlled by a Markov-chain dynamics. GPK processes possess trajectory regularity (almost everywhere) and asymptotic Kac limit, namely the convergence towards Brownian motion (and to stochastic dynamics driven by Wiener perturbations), which characterizes also the long-term/long-distance properties of these processes. In this article we introduce the structural properties of GPK processes, leaving all the physical implications to part II and part III.