Passage times of fast inhomogeneous immigration processes

Abstract

In many biophysical systems, key events are triggered when the fastest of many random searchers find a target. Most mathematical models of such systems assume that all searchers are initially present in the search domain, which permits the use of classical extreme value theory. In this paper, we explore $k$th passage times of inhomogeneous immigration processes where searchers are added to the domain over time either through time inhomogeneous rates or a Yule (pure birth) process. We rigorously prove convergence in distribution and convergence of moments of the $k$th passage times for both processes as immigration rates grow. In particular, we relate immigration with time inhomogeneous rates to previous work where all searchers are initially present through a coupling argument and demonstrate how immigration through a Yule process can be viewed as a time inhomogeneous immigration process with a random time shift. For Yule immigration, we find that the extreme distributions depart from the classical family of Frechet, Gumbel, and Weibull, and we compare our results to classical theorems on branching Brownian motion. This work offers one of the few examples where extreme value distributions can be obtained exactly for random variables which are neither independent nor identically distributed.