# First passage of a particle in a potential under stochastic resetting: a vanishing transition of optimal resetting rate

## TL;DR

This paper investigates how a strong external potential causes the optimal stochastic resetting rate to vanish in a diffusing particle's first passage time, revealing a continuous transition with universal characteristics.

## Contribution

It identifies a continuous transition where the optimal resetting rate disappears under strong potentials and derives the universal power-law behavior near the critical point.

## Key findings

- Optimal resetting rate vanishes at a critical potential strength.
- The transition is continuous and characterized by a universal power-law exponent.
- The power-law behavior is confirmed analytically and numerically.

## Abstract

First passage in a stochastic process may be influenced by the presence of an external confining potential, as well as "stochastic resetting" in which the process is repeatedly reset back to its initial position. Here we study the interplay between these two strategies, for a diffusing particle in an one-dimensional trapping potential $V(x)$, being randomly reset at a constant rate $r$. Stochastic resetting has been of great interest as it is known to provide an "optimal rate" ($r_*$) at which the mean first passage time is a minimum. On the other hand an attractive potential also assists in first capture process. Interestingly, we find that for a sufficiently strong external potential, the advantageous optimal resetting rate vanishes (i.e. $r_*\to 0$). We derive a condition for this $optimal$ $resetting$ $rate$ $vanishing$ $transition$, which is continuous. We study this problem for various functional forms of $V(x)$, some analytically, and the rest numerically. We find that the optimal rate $r_*$ vanishes with the deviation from critical strength of the potential as a power law with an exponent $\beta$ which appears to be universal.