Continuous-time random walks with reset events: Historical background and new perspectives

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.

11 pages, 5 figures

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• Stochastic resetting on comb-like structures
• Anomalous diffusion in random-walks with memory-induced relocations
• Fat Tails and Black Swans: Exact Results for Multiplicative Processes with Resets
• Non-homogeneous random walks with stochastic resetting: an application to the Gillis model
• The one-dimensional telegraphic process with noninstantaneous stochastic resetting