Monotonous continuous-time random walks with drift and stochastic reset events

TL;DR

This paper analyzes monotonous continuous-time random walks with drift and stochastic resets, deriving statistical properties such as stationary distributions, power-law behaviors, and exit times, supported by Monte Carlo simulations.

Contribution

It introduces a comprehensive analysis of monotonous CTRWs with drift and resets, deriving explicit formulas for key statistics and demonstrating power-law behaviors.

Findings

Existence of stationary transition probability density for any drift strength

Derivation of formulas for survival probability and mean exit time

Monte Carlo simulations confirm analytical results

Abstract

In this paper we consider a stochastic process that may experience random reset events which bring suddenly the system to the starting value and analyze the relevant statistical magnitudes. We focus our attention on monotonous continuous-time random walks with a constant drift: the process increases between the reset events, either by the effect of the random jumps, or by the action of the deterministic drift. As a result of all these combined factors interesting properties emerge, like the existence|for any drift strength|of a stationary transition probability density function, or the faculty of the model to reproduce power-law-like behavior. General formulas for two extreme statistics, the survival probability and the mean exit time, are also derived. To corroborate in an independent way the results of the paper, Monte Carlo methods were used. These numerical estimations are in full agreement with the analytical predictions.