The time it takes the fastest searcher out of N ≫ 1 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.

N中最快的搜索者花费的时间≫ 1 搜索者找到一个目标决定了许多物理、化学和生物过程的时间尺度。该时间称为极端首次通过时间 (FPT)，通常比单个搜索者的 FPT 快得多。扩散的极端 FPT 已经研究了几十年，但对其他类型的随机过程知之甚少。在本文中，我们研究了分段确定性马尔可夫过程 (PDMP) 的极端 FPT 的分布。PDMP 是一大类随机过程，它们在随机事件之间确定性地演化。使用经典的极值理论，我们证明了基于单个搜索器的 FPT 的短时间分布，在许多搜索器的限制下产生极值 FPT 的分布和矩的一般定理。然后我们将这些定理应用于一些规范的 PDMP，包括一、二和三个空间维度的奔跑和翻滚搜索器。我们在一些生物系统的背景下讨论了我们的结果，并展示了我们的方法如何解释扩散的非物理特性，这对于极端统计来说可能是有问题的。