First-passage time

The first-passage time is the earliest time a stochastic process reaches a specified state or set of states. Its distribution describes when that first arrival occurs across repeated realizations.

Survival analysis: The study of time until an event occurs, accounting for observations whose event time is not yet known. Survival probabilities track the process remaining short of its target before first arrival.
Stochastic process: A collection of random variables indexed by time or another parameter, representing the evolution of a random system. A first-passage time is defined by when such a process reaches a target.
First-passage process in chemical kinetics: A stochastic description of the time until molecules first meet, react, or reach a reaction site. Reaction rates can depend on how quickly diffusing molecules first encounter one another.
Recurrence time: The time required for a dynamical or stochastic system to return to a specified state or region. First passage starts from an initial state; recurrence begins after departure from the target.
Absorbing state: A state that, once entered, cannot be left in a stochastic process. Making the target absorbing converts first arrival into permanent capture.
Stopping time: A random time whose occurrence by any given time can be determined from information available up to then. First passage is a stopping time when target arrival is observable from the process history.
Narrow escape problem: The problem of estimating how long a diffusing particle takes to find a small target on a confining boundary. It quantifies first arrival at tiny openings in cells and other bounded domains.
Mean recurrence time: The expected interval between successive visits to a state or region. It averages repeated returns rather than the initial wait for a target's first arrival.
Master equation: A differential equation describing how probabilities over the states of a stochastic system change over time. Its probability flow can determine the rate at which trajectories reach a target.
Markov property: The property that future evolution depends on the present state, not on the full past, once the present is known. Markov structure often makes first-passage calculations tractable.

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