Non-standard diffusion under Markovian resetting in bounded domains

TL;DR

This paper studies how Markovian resetting affects a one-dimensional random walk with general waiting times in bounded domains, revealing conditions for optimal resetting to minimize exit times.

Contribution

It generalizes previous Markovian results to non-Markovian waiting times, identifying how the variability of waiting times influences the benefit of resetting.

Findings

Resetting can be never beneficial, beneficial depending on reset position, or always beneficial.

The benefit of resetting depends on the relative standard deviation of waiting times.

Optimal reset rate minimizes mean exit passage time under certain conditions.

Abstract

We consider a walker moving in a one-dimensional interval with absorbing boundaries under the effect of Markovian resettings to the initial position. The walker's motion follows a random walk characterized by a general waiting time distribution between consecutive short jumps. We investigate the existence of an optimal reset rate, which minimizes the mean exit passage time, in terms of the statistical properties of the waiting time probability. Generalizing previous results restricted to Markovian random walks, we here find that, depending on the value of the relative standard deviation of the waiting time probability, resetting can be either (i) never beneficial, (ii) beneficial depending on the distance of the reset to the boundary, or (iii) always beneficial.