Extreme first passage times of piecewise deterministic Markov processes

TL;DR

This paper analyzes the distribution of extreme first passage times for piecewise deterministic Markov processes, providing theoretical results and applications to biological search models, extending understanding beyond diffusion processes.

Contribution

It introduces general theorems for extreme FPTs of PDMPs using classical extreme value theory, applicable to various biological search scenarios.

Findings

Derived distribution and moments of extreme FPTs for PDMPs

Applied theorems to run and tumble search models in multiple dimensions

Addressed limitations of diffusion models in extreme statistics

Abstract

The time it takes the fastest searcher out of $N\gg1$ searchers to find a target determines the timescale of many physical, chemical, and biological processes. This time is called an extreme first passage time (FPT) and is typically much faster than the FPT of a single searcher. Extreme FPTs of diffusion have been studied for decades, but little is known for other types of stochastic processes. In this paper, we study the distribution of extreme FPTs of piecewise deterministic Markov processes (PDMPs). PDMPs are a broad class of stochastic processes that evolve deterministically between random events. Using classical extreme value theory, we prove general theorems which yield the distribution and moments of extreme FPTs in the limit of many searchers based on the short time distribution of the FPT of a single searcher. We then apply these theorems to some canonical PDMPs, including run and tumble searchers in one, two, and three space dimensions. We discuss our results in the context of some biological systems and show how our approach accounts for an unphysical property of diffusion which can be problematic for extreme statistics.