Editing Wiener process (section)

== Characterisations of the Wiener process ==
The Wiener process ''<math>W_t</math>'' is characterised by the following properties:<ref>{{cite book |last=Durrett |first=Rick |author-link=Rick Durrett |date=2019 |title=Probability: Theory and Examples |edition=5th |chapter=Brownian Motion |publisher=Cambridge University Press |isbn=9781108591034}}</ref>

#<math>W_0= 0</math> [[almost surely]]
#<math>W</math> has [[independent increments]]: for every <math>t>0,</math> the future increments <math>W_{t+u} - W_t,</math> <math>u \ge 0,</math> are independent of the past values <math>W_s</math>, <math>s< t.</math> 
#<math>W</math> has Gaussian increments: <math>W_{t+u} - W_t</math> is normally distributed with mean <math>0</math> and variance <math>u</math>, <math>W_{t+u} - W_t\sim \mathcal N(0,u).</math>
#<math>W</math> has almost surely continuous paths: <math>W_t</math> is almost surely continuous in <math>t</math>.

That the process has independent increments means that if {{math|0 ≤ ''s''<sub>1</sub> < ''t''<sub>1</sub> ≤ ''s''<sub>2</sub> < ''t''<sub>2</sub>}} then {{math|''W''<sub>''t''<sub>1</sub></sub> − ''W''<sub>''s''<sub>1</sub></sub>}} and {{math|''W''<sub>''t''<sub>2</sub></sub> − ''W''<sub>''s''<sub>2</sub></sub>}} are independent random variables, and the similar condition holds for ''n'' increments.

An alternative characterisation of the Wiener process is the so-called ''Lévy characterisation'' that says that the Wiener process is an almost surely continuous [[martingale (probability theory)|martingale]] with {{math|1=''W''<sub>0</sub> = 0}} and [[quadratic variation]] {{math|1=[''W''<sub>''t''</sub>, ''W''<sub>''t''</sub>] = ''t''}} (which means that {{math|''W''<sub>''t''</sub><sup>2</sup> − ''t''}} is also a martingale).

A third characterisation is that the Wiener process has a spectral representation as a sine series whose coefficients are independent ''N''(0, 1) random variables.  This representation can be obtained using the [[Karhunen–Loève theorem]].

Another characterisation of a Wiener process is the [[definite integral]] (from time zero to time ''t'') of a zero mean, unit variance, delta correlated ("white") [[Gaussian process]].<ref>{{Cite journal|last1=Huang|first1=Steel T.| last2=Cambanis|first2=Stamatis| date=1978|title=Stochastic and Multiple Wiener Integrals for Gaussian Processes|journal=The Annals of Probability|volume=6|issue=4|pages=585–614|doi=10.1214/aop/1176995480 |jstor=2243125 |issn=0091-1798|doi-access=free}}</ref>

The Wiener process can be constructed as the [[scaling limit]] of a [[random walk]], or other discrete-time stochastic processes with stationary independent increments. This is known as [[Donsker's theorem]]. Like the random walk, the Wiener process is recurrent in one or two dimensions (meaning that it returns almost surely to any fixed [[neighborhood (mathematics)|neighborhood]] of the origin infinitely often) whereas it is not recurrent in dimensions three and higher (where a multidimensional Wiener process is a process such that its coordinates are independent Wiener processes).<ref>{{cite web |title= Pólya's Random Walk Constants |website= Wolfram Mathworld| url = https://mathworld.wolfram.com/PolyasRandomWalkConstants.html}}</ref> Unlike the random walk, it is [[scale invariance|scale invariant]], meaning that
<math display="block">\alpha^{-1} W_{\alpha^2 t}</math>
is a Wiener process for any nonzero constant {{mvar|α}}. The '''Wiener measure''' is the [[Law (stochastic processes)|probability law]] on the space of [[continuous function]]s {{math|''g''}}, with {{math|1=''g''(0) = 0}}, induced by the Wiener process. An [[integral]] based on Wiener measure may be called a '''Wiener integral'''.