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A Poisson process N(t) with rate λ counts events occurring randomly in time: it has independent and stationary increments, with N(t) - N(s) ~ Poisson(λ(t-s)) for t > s. Equivalently, inter-arrival times are i.i.d. Exponential(λ). It is the continuous-time counting process with the "memoryless" property — the process restarts probabilistically after each event. The Poisson process is the fundamental model for random arrivals and the jump-process counterpart of Brownian motion.
The Poisson process is the simplest and most important continuous-time counting process. It counts events (arrivals, failures, transactions) that occur randomly in time at a constant average rate λ. The three equivalent characterizations are: (1) N(t) has independent, stationary Poisson-distributed increments with N(t)-N(s) ~ Poisson(λ(t-s)); (2) inter-arrival times T₁, T₂, ... are i.i.d. Exponential(λ); (3) for infinitesimally small dt, P(one event in [t, t+dt]) = λdt + o(dt) and P(two or more events) = o(dt). Each characterization captures a different aspect — the distribution of counts, the distribution of waiting times, and the infinitesimal event rate.
The memoryless property of the exponential distribution is the Poisson process's signature. Given that no event has occurred for s time units, the distribution of the remaining waiting time is the same Exponential(λ) — the process "doesn't remember" how long it has been waiting. This is the unique continuous distribution with this property (the geometric distribution is the discrete analogue). The memoryless property implies that the Poisson process is Markov: the future depends only on the current count, not on the history of arrival times.
Key structural properties include superposition (merging independent Poisson processes produces another Poisson process with summed rates), thinning (independently keeping each event with probability p produces a Poisson process with rate λp), and the order-statistic property (given N(t) = n, the n arrival times are distributed as the order statistics of n i.i.d. Uniform([0,t]) variables). The compound Poisson process X(t) = Σᵢ₌₁^{N(t)} Yᵢ attaches an i.i.d. random magnitude Yᵢ to each event, modeling aggregate claims in insurance, total demand, or cumulative price jumps.
The Poisson process is the jump-process counterpart of Brownian motion. Where Brownian motion has continuous paths and is the building block for continuous stochastic calculus, the Poisson process has piecewise-constant paths with unit jumps and is the building block for jump-process calculus. The compensated Poisson process Ñ(t) = N(t) - λt is a martingale — the counting process minus its expected rate. This martingale plays the same role for jump processes that Brownian motion plays for continuous processes, and it is the starting point for the stochastic calculus of jump processes and Levy processes.