# Median and Mode in First Passage under Restart

## TL;DR

This paper investigates how restart strategies influence the median and mode of first-passage times, providing criteria for improvements and demonstrating the benefits of non-uniform restart strategies through theoretical analysis and examples.

## Contribution

It introduces a general criterion for when restart reduces the median and shows that restart always decreases the mode, with optimization strategies applicable to various processes.

## Key findings

- Restart can lower the median first-passage time under certain conditions.
- Restart always reduces the mode of the first-passage time distribution.
- Non-uniform restart strategies can optimize mean and median first-passage times.

## Abstract

Restart -- interrupting a stochastic process followed by a new start -- is known to improve the mean time to its completion, and the general conditions under which such an improvement is achieved are now well understood. Here, we explore how restart affects other important metrics of first-passage phenomena, namely the median and the mode of the first-passage time distribution. Our analysis provides a general criterion for when restart lowers the median time, and demonstrates that restarting is always helpful in reducing the mode. Additionally, we show that simple non-uniform restart strategies allow to optimize the mean and the median first-passage times, regardless of the characteristic time scales of the underlying process. These findings are illustrated with the canonical example of a diffusive search with resetting.