A First Look at First-Passage Processes

TL;DR

This work offers a concise, tutorial-like survey of first-passage processes, defining the core quantities $F(oldsymbol{r},t)$ and $P(oldsymbol{r},t)$ and showing how to derive first-passage information from occupation data via generating functions or Laplace transforms. It then develops exact or scalable results for canonical geometries (half-line and finite interval) and highlights a powerful electrostatic analogy that turns time-dependent diffusion questions into static electrostatic problems. The author surveys a suite of applications—ranging from hitting probabilities for spheres and wedges to receptor-uptake on cell surfaces, predator-prey-like stalking problems, expanding domains, and birth–death kinetics—demonstrating the versatility of first-passage methods and a diverse toolbox (Green's functions, image methods, conformal mappings, Weyl chambers, and scaling analyses). The work underscores both the mathematical elegance and practical relevance of first-passage theory in physics, chemistry, and biology, and provides accessible pathways for applying these ideas to complex stochastic systems.

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.

Figures (10)

• Figure 1: Schematic diagrammatic relation between the occupation probability of a random walk (whose propagation is represented by a wavy line) and the first-passage probability (straight line).
• Figure 2: Concentration profile of a diffusing particle on the absorbing infinite half line $(0,\infty)$ at $Dt=10$, with $x_0=2$ (solid curve). Also shown are the component Gaussian (blue dashed) and image anti-Gaussian (red dot-dash) and their superposition in the physical region $x>0$.
• Figure 3: Schematic decomposition of a random walk path from $x_0$ to $L$ into the outcome after one step (red) and the remainder from $x'$ to $L$. The factors $1/2$ account for the probabilities associated with the first step of the decomposed paths.
• Figure 4: Dependence of the exit probability to $x=L$ on $u_0=x_0/L$ in the interval $[0,L]$ for various Péclet numbers $P\!e$.
• Figure 5: Unconditional and conditional average exit times from the finite interval $[0,L]$, normalized by $L^2/2D$, as a function of the dimensionless initial position $x_0/L$.