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

TL;DR

This paper studies a one-dimensional continuous-time random walk with random reset events, deriving formulas for key properties and verifying results through numerical simulations, offering new insights into systems with stochastic resets.

Contribution

It introduces a comprehensive analysis of a monotonic random walk with resets, providing explicit formulas for propagator, survival probability, and mean first-passage time, with validation.

Findings

Derived general formulas for propagator, survival probability, and mean first-passage time.

Validated analytical results through numerical simulations.

Revealed interesting properties emerging from the interplay of drift, jumps, and resets.

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.