A First Look at First-Passage Processes

TL;DR

This paper introduces the fundamental concepts of first-passage processes, explores their mathematical properties, and discusses various applications across physics and biology, providing foundational insights into stochastic boundary-crossing phenomena.

Contribution

It offers a comprehensive overview of first-passage theory, connecting it to electrostatics and illustrating its applications in diverse scientific contexts.

Findings

Connection between first-passage probability and occupation probability

Analysis of first-passage on finite and semi-infinite domains

Applications to reaction rates, cell receptors, and stochastic processes

Abstract

These notes are based on the lectures that I gave (virtually) at the Bruneck Summer School in 2021 on first-passage processes and some applications of the basic theory. I begin by defining what is a first-passage process and presenting the connection between the first-passage probability and the familiar occupation probability. Some basic features of first passage on the semi-infinite line and a finite interval are then discussed, such as splitting probabilities and first-passage times. I also treat the fundamental connection between first passage and electrostatics. A number of applications of first-passage processes are then presented, including the hitting probability for a sphere in greater than two dimensions, reaction rate theory and its extension to receptors on a cell surface, first-passage inside an infinite absorbing wedge in two dimensions, stochastic hunting processes in one dimension, the survival of a diffusing particle in an expanding interval, and finally the dynamics of the classic birth-death process.