[Paper Review] A First Look at First-Passage Processes

S. Redner|arXiv (Cornell University)|Jan 25, 2022

Diffusion and Search Dynamics6 citations

TL;DR

This paper provides a comprehensive introduction to first-passage processes in stochastic dynamics, deriving the fundamental link between first-passage and occupation probabilities via generating functions. It establishes key results for diffusion in confined geometries—such as the half-line, finite intervals, and absorbing wedges—and applies them to physical and biological systems, including reaction rates, neuronal firing, and population extinction, revealing universal scaling laws like survival probability decaying as $ t^{-1} $ in birth-death processes and $ t^{-1/2} $ in diffusion.

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.

Motivation & Objective

• To establish the theoretical framework connecting first-passage probabilities with occupation probabilities using generating functions.
• To analyze first-passage phenomena in confined geometries, such as the half-line and finite intervals, focusing on splitting probabilities and first-passage times.
• To explore the electrostatic analogy in first-passage processes, enabling analytical solutions for complex boundary conditions.
• To apply first-passage theory to real-world systems, including reaction kinetics on cell surfaces, stochastic hunting, and population dynamics.
• To demonstrate the universality of first-passage scaling laws across different stochastic processes, such as $ S(t) \sim t^{-1} $ in birth-death processes versus $ S(t) \sim t^{-1/2} $ in diffusion.

Proposed method

• Derive the convolution relation between occupation probability $ P(\mathbf{r},t) $ and first-passage probability $ F(\mathbf{r},t) $, expressed as $ P(\mathbf{r},t) = \delta_{\mathbf{r},0}\delta_{t,0} + \sum_{t' \leq t} F(\mathbf{r},t') P(0,t-t') $.
• Use generating functions $ P(\mathbf{r},z) $ and $ F(\mathbf{r},z) $ to transform the convolution into an algebraic relation: $ F(\mathbf{r},z) = P(\mathbf{r},z)/P(0,z) $ for $ \mathbf{r} \neq 0 $.
• Apply Laplace transforms to the diffusion equation with absorbing boundaries to compute first-passage time distributions in continuous space and time.
• Leverage the electrostatic analogy, mapping first-passage problems to potential theory via the Green’s function and flux at absorbing boundaries.
• Solve the master equation for the birth-death process using a generating function $ g(z,t) = \sum_n P_n(t) z^n $, transforming it into a first-order PDE: $ g_t = (1-z)^2 g_z $.
• Introduce a change of variables $ y = 1/(1-z) $ to reduce the PDE to the wave equation $ g_t = g_y $, solvable by the method of characteristics.

Experimental results

Research questions

• RQ1 What is the exact relationship between first-passage and occupation probabilities in random walks?
• RQ2 How do first-passage times and splitting probabilities behave on a semi-infinite line and a finite interval?
• RQ3 What is the connection between first-passage processes and electrostatics, and how can it be used to solve boundary value problems?
• RQ4 How does the survival probability of a diffusing particle decay in time when confined in an expanding interval?
• RQ5 What are the first-passage properties of the birth-death process, and how do they compare to those of simple diffusion?

Key findings

• The first-passage probability $ F(\mathbf{r},z) $ is fully determined by the occupation probability $ P(\mathbf{r},z) $ through the relation $ F(\mathbf{r},z) = P(\mathbf{r},z)/P(0,z) $ for $ \mathbf{r} \neq 0 $.
• On the half-line, the first-passage time distribution for a particle starting at $ x_0 > 0 $ is derived via Laplace transform of the diffusion equation with an absorbing boundary at $ x=0 $.
• For a diffusing particle in a growing interval, the survival probability decays as $ t^{-1/2} $, similar to standard diffusion, but with modified time-dependent boundary conditions.
• In the birth-death process with linear birth rate, the survival probability $ S(t) = 1 - P_0(t) $ decays as $ 1/(1+t) $, implying infinite mean extinction time despite sure extinction.
• The generating function solution for the birth-death process yields exact expressions: $ P_0(t) = t/(1+t) $, $ P_n(t) = t^{n-1}/(1+t)^{n+1} $ for $ n \geq 1 $.
• The electrostatic analogy allows mapping first-passage problems to potential theory, enabling exact solutions for complex geometries like absorbing wedges and spherical receptors.