Exact moments in a continuous time random walk with complete memory of its history

2006/03/31 by Francis N. C. Paraan, Jose Perico Esguerra, J. P. Esguerra · 6 citations
Biochemistry, Genetics and Molecular Biology · Physics and Astronomy · #Diffusion and Search Dynamics #Spectroscopy and Quantum Chemical Studies #Theoretical and Computational Physics #cond-mat.stat-mech

paper · pdf · doi:10.1103/physreve.74.032101

published as Phys. Rev. E 74, 032101 (2006) · Revtex4, 10 pages, 2 figures. v2: applications discussed, clarity improved, corrected scaling of third moment

arxiv created 2006/07/03 · openalex publication_date 2006/09/25 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We present a continuous time generalization of a random walk with complete memory of its history [Phys. Rev. E 70, 045101(R) (2004)] and derive exact expressions for the first four moments of the distribution of displacement when the number of steps is Poisson distributed. We analyze the asymptotic behavior of the normalized third and fourth cumulants and identify new transitions in a parameter regime where the random walk exhibits superdiffusion. These transitions, which are also present in the discrete time case, arise from the memory of the process and are not reproduced by Fokker-Planck approximations to the evolution equation of this random walk.

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