From persistent random walks to the telegraph noise

2008/10/03 by Samuel Herrmann, Herrmann, Samuel, Pierre Vallois +1
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #Diffusion and Search Dynamics #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #Topological and Geometric Data Analysis #math.PR

paper · pdf · doi:10.48550/arxiv.0810.0650

arxiv created 2008/10/03 · openalex publication_date 2008/10/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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≥ 0).

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