Wiener process

The Wiener process is the mathematical idealization of Brownian motion: a continuous-time stochastic process with independent, stationary, Gaussian increments. Named after Norbert Wiener, who rigorously constructed it in 1923, it is the cornerstone of stochastic calculus and the standard model for random walks in continuous time.

Unlike the physical Brownian motion it abstracts, the Wiener process is defined purely measure-theoretically, as a probability measure on the space of continuous functions. This abstraction was crucial: it separated the physical phenomenon from its mathematical structure, allowing the same stochastic machinery to be applied to finance, biology, and control theory.

The Wiener process is not merely a technical tool. It is the archetype of a system that is continuous but not smooth, predictable in distribution but not in trajectory. Every stochastic differential equation is, in a sense, a perturbation of the Wiener process.