Editing Wiener process (section)

{{Use American English|date=January 2019}}
{{Short description|Stochastic process generalizing Brownian motion}}
{{More footnotes|date=February 2010}}
{{Infobox probability distribution|name=Wiener Process|pdf_image=Wiener process with sigma.svg|mean=<math> 0 </math>|variance=<math>\sigma^2 t</math>|type=multivariate}}[[File:wiener process zoom.png|thumb|300px|A single realization of a one-dimensional Wiener process]]
[[File:WienerProcess3D.svg|thumb|300px|A single realization of a three-dimensional Wiener process]]

In [[mathematics]], the '''Wiener process''' (or '''Brownian motion''', due to its historical connection with [[Brownian motion|the physical process of the same name]]) is a real-valued [[continuous-time]] [[stochastic process]] discovered by [[Norbert Wiener]].<ref>{{cite book |last= Dobrow|first=Robert |author-link=Robert Dobrow |date=2016 |title=Introduction to Stochastic Processes with R |publisher=Wiley |pages=321–322 |doi=10.1002/9781118740712 |bibcode=2016ispr.book.....D |url=https://onlinelibrary.wiley.com/doi/book/10.1002/9781118740712 |isbn=9781118740651}}</ref><ref>N.Wiener Collected Works vol.1</ref> It is one of the best known [[Lévy process]]es ([[càdlàg]] stochastic processes with [[stationary increments|stationary]] [[independent increments]]). It occurs frequently in pure and [[applied mathematics]], [[economy|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 [[martingale (probability theory)|martingale]]s. 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 process]]es 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 [[Filter (signal processing)|filtering theory]] and disturbances in [[control theory]].

The Wiener process has applications throughout the mathematical sciences. In physics it is used to study Brownian motion and other types of diffusion via the [[Fokker–Planck equation|Fokker–Planck]] and [[Langevin equation]]s. It also forms the basis for the rigorous [[path integral formulation]] of [[quantum mechanics]] (by the [[Feynman–Kac formula]], a solution to the [[Schrödinger equation]] can be represented in terms of the Wiener process) and the study of [[eternal inflation]] in [[physical cosmology]]. It is also prominent in the [[mathematical finance|mathematical theory of finance]], in particular the [[Black–Scholes]] option pricing model.<ref>Shreve and Karatsas</ref>