First passage times with fast immigration

TL;DR

This paper analyzes the distribution of the fastest first passage times in systems where searchers enter a domain at a constant rate, providing exact extreme value statistics for correlated stochastic processes.

Contribution

It introduces a model with progressive searcher entry and derives exact distributions for the fastest FPTs in correlated stochastic systems, extending extreme value theory.

Findings

Derived probability distribution and moments of the $k$-th fastest FPT.

Applicable to random walks on networks and diffusion in continuous spaces.

Exact extreme value statistics for strongly correlated variables.

Abstract

Many scientific questions can be framed as asking for a first passage time (FPT), which generically describes the time it takes a random "searcher" to find a "target." The important timescale in a variety of biophysical systems is the time it takes the fastest searcher(s) to find a target out of many searchers. Previous work on such fastest FPTs assumes that all searchers are initially present in the domain, which makes the problem amenable to extreme value theory. In this paper, we consider an alternative model in which searchers progressively enter the domain at a constant "immigration" rate. In the fast immigration rate limit, we determine the probability distribution and moments of the $k$-th fastest FPT. Our rigorous theory applies to many models of stochastic motion, including random walks on discrete networks and diffusion on continuous state spaces. Mathematically, our analysis involves studying the extrema of an infinite sequence of random variables which are both not independent and not identically distributed. Our results constitute a rare instance in which extreme value statistics can be determined exactly for strongly correlated random variables.