Eur. Phys. J. B (2017) 90: 176
https://doi.org/10.1140/epjb/e2017-80348-4

Regular Article

Continuous-time random walks with reset events^*

Historical background and new perspectives

¹ Departament de Física de la Matèria Condensada, Universitat de Barcelona (UB), Martí i Franquès 1, 08028 Barcelona, Spain
² Universitat de Barcelona Institute of Complex Systems (UBICS), Universitat de Barcelona, Barcelona, Spain
³ Departamento de Matemáticas & Instituto Universitario de Física Fundamental y Matemáticas, Universidad de Salamanca, Plaza Merced s/n, 37008 Salamanca, Spain

^a e-mail: miquel.montero@ub.edu

Received: 15 June 2017
Received in final form: 4 July 2017
Published online: 25 September 2017

Abstract

In this paper, we consider a stochastic process that may experience random reset events which relocate the system to its starting position. We focus our attention on a one-dimensional, monotonic continuous-time random walk with a constant drift: the process moves in a fixed direction between the reset events, either by the effect of the random jumps, or by the action of a deterministic bias. However, the orientation of its motion is randomly determined after each restart. As a result of these alternating dynamics, interesting properties do emerge. General formulas for the propagator as well as for two extreme statistics, the survival probability and the mean first-passage time, are also derived. The rigor of these analytical results is verified by numerical estimations, for particular but illuminating examples.