From persistent random walks to the telegraph noise

TL;DR

This paper investigates a class of memory-influenced persistent random walks, demonstrating their convergence to non-Markovian processes that generalize Brownian motion, and explores their relation to the telegraph equation.

Contribution

It introduces a new family of limit processes for persistent random walks, including a non-Markovian process linked to the telegraph equation, expanding understanding of such stochastic models.

Findings

Weak convergence of rescaled walks to non-Markov processes

Representation of the telegraph process via counting processes

Detailed analysis of the Markov process (Z_t, N_t)

Abstract

We study a family of memory-based persistent random walks and we prove weak convergences after space-time rescaling. The limit processes are not only Brownian motions with drift. We have obtained a continuous but non-Markov process $(Z_t)$ which can be easely expressed in terms of a counting process $(N_t)$. In a particular case the counting process is a Poisson process, and $(Z_t)$ permits to represent the solution of the telegraph equation. We study in detail the Markov process $((Z_t,N_t); t\ge 0)$.