Restarts delay escape over a potential barrier

TL;DR

This paper investigates how restarts, especially position-dependent ones, delay the escape of a stochastic searcher over a potential barrier, and explores modifications to reduce this delay.

Contribution

It demonstrates that restarts universally delay escape times, introduces a model with position-dependent restart rates, and suggests methods to mitigate the delay.

Findings

Restarts delay escape times regardless of restart distribution or location.

Position-dependent restart rates further increase the delay compared to classical Kramers escape.

Modifications to time overheads can help expedite escape under restart strategies.

Abstract

In the barrier escape problem, a random searcher starting at the energy minima tries to escape the barrier under the effect of thermal fluctuations. If the random searcher is subject to successive restarts at the bottom of the well, then its escape over the barrier top is delayed compared to the time it would take in absence of restarts. When restarting at an intermediate location, the time required by the random searcher to go from the bottom of the well to the restart location should be considered. Taking into account this time overhead, we find that restarts delay escape, independent of the specific nature of the distribution of restart times, or the location of restart, or the specific details of the random searcher. For the special case of Poisson restarts, we study the escape problem for a Brownian particle with a position-dependent restart rate $r(x)\theta(x^p_0-x)$, with $x^p_0$ being the location of restart. We find that position-dependent restarts delay the escape as compared to Kramers escape time. We also study ways of modifying time overheads which help expedite escape under restarts.