Wiener process

Wiener Process
Probability density function
Mean	0
Variance	${\displaystyle \sigma ^{2}t}$

In mathematics, the Wiener process (or Brownian motion, due to its historical connection with the physical process of the same name) is a real-valued continuous-time stochastic process named after Norbert Wiener. It is one of the best known Lévy processes (càdlàg stochastic processes with stationary independent increments). It occurs frequently in pure and applied mathematics, economics, quantitative finance, evolutionary biology, and physics.

The Wiener process plays an important role in both pure and applied mathematics. In pure mathematics, the Wiener process gave rise to the study of continuous time martingales. It is a key process in terms of which more complicated stochastic processes can be described. As such, it plays a vital role in stochastic calculus, diffusion processes and even potential theory. It is the driving process of Schramm–Loewner evolution. In applied mathematics, the Wiener process is used to represent the integral of a white noise Gaussian process, and so is useful as a model of noise in electronics engineering (see Brownian noise), instrument errors in filtering theory and disturbances in control theory.

The Wiener process has applications throughout the mathematical sciences. In physics, researchers use it to model Brownian motion and other types of diffusion, often through the Fokker–Planck and Langevin equations, which describe how random motion evolves over time. It also underpins the rigorous path integral formulation of quantum mechanics: by the Feynman–Kac formula, one can represent solutions to the Schrödinger equation in terms of the Wiener process. In physical cosmology, it also appears in models of eternal inflation. The Wiener process is prominent in the mathematical theory of finance as well, in particular the Black–Scholes option pricing model.