firstpassage

Firstpassage, commonly written as first-passage or first-passage time (FPT), refers to a concept in probability theory and stochastic processes describing the time it takes for a stochastic process to reach a predefined state or boundary for the first time. For a process X(t) with X(0) = x and a target value a, the first passage time to a is defined as T_a = inf{ t ≥ 0 : X(t) = a }. In higher dimensions, the target is typically a boundary or a domain boundary ∂D.

In one-dimensional diffusion and Brownian motion, the first passage time is a fundamental quantity used to characterize how long it takes for random fluctuations to hit a threshold. Different processes yield different distributions for T_a: standard Brownian motion without drift has a Lévy distribution for T_a, while Brownian motion with drift μ gives an inverse Gaussian distribution. For general diffusion processes, the first passage problem is often formulated through backward and forward equations such as the Kolmogorov or Fokker–Planck equations, with appropriate absorbing boundary conditions. Solutions are sometimes obtained analytically for simple geometries or numerically via simulation of sample paths or by solving boundary value problems.

Applications of first passage times span many fields. In physics and chemistry, they model reaction rates and diffusion-controlled processes. In neuroscience, they describe the time for a membrane potential to reach a firing threshold. In finance, first passage concepts underpin pricing and risk features of barrier options and credit events. In random media and ecology, they are used to study escape and travel times in heterogeneous environments. A related concept is first passage percolation, a growth model that focuses on minimal travel times across random media.