Probability distributions for the run-and-tumble models with variable speed and tumbling rate

Angelani, Luca; Garra, Roberto

doi:10.15559/18-VMSTA127

Modern Stochastics: Theory and Applications

Probability distributions for the run-and-tumble models with variable speed and tumbling rate

Volume 6, Issue 1 (2019), pp. 3–12

Luca Angelani Roberto Garra

https://doi.org/10.15559/18-VMSTA127

Pub. online: 21 December 2018 Type: Research Article

Open Access

Received
16 July 2018

Revised
8 November 2018

Accepted
7 December 2018

Published
21 December 2018

Abstract

In this paper we consider a telegraph equation with time-dependent coefficients, governing the persistent random walk of a particle moving on the line with a time-varying velocity $c(t)$ and changing direction at instants distributed according to a non-stationary Poisson distribution with rate $\lambda (t)$. We show that, under suitable assumptions, we are able to find the exact form of the probability distribution. We also consider the space-fractional counterpart of this model, finding the characteristic function of the related process. A conclusive discussion is devoted to the potential applications to run-and-tumble models.

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Keywords

Telegraph equation with time-dependent velocity run-and-tumble models exact marginal probability distribution

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