Fastest first-passage time statistics for time-dependent particle injection

A common scenario in a variety of biological systems is that multiple particles are searching in parallel for an immobile target located in a bounded domain, and the fastest among them that arrives to the target first triggers a given desirable or detrimental process. The statistics of such extreme events -- the \textit{fastest\/} first-passage to the target -- is well-understood by now through a series of theoretical analyses, but exclusively under the assumption that all $N$ particles start \textit{simultaneously\/}, i.e., all are introduced into the domain instantly, by $\delta$-function-like pulses. However, in many practically important situations this is not the case: in order to start their search, the particles often have to enter first into a bounded domain, e.g., a cell or its nucleus, penetrating through gated channels or nuclear pores. This entrance process has a random duration so that the particles appear in the domain sequentially and with a time delay. Here we focus on the effect of such an extended-in-time injection of multiple particles on the fastest first-passage time (fFPT) and its statistics. We derive the full probability density function $H_N(t)$ of the fFPT with an arbitrary time-dependent injection intensity of $N$ particles. Under rather general assumptions on the survival probability of a single particle and on the injection intensity, we derive the large-$N$ asymptotic formula for the mean fFPT, which is quite different from that obtained for the instantaneous $\delta$-pulse injection. The extended injection is also shown to considerably slow down the convergence of $H_N(t)$ to the large-$N$ limit -- the Gumbel distribution -- so that the latter may be inapplicable in the most relevant settings with few tens to few thousands of particles.