Editing Brownian motion (section)

==Mathematics==
{{Main|Wiener process}}
[[File:2D Random Walk 400x400.ogv|thumb|right|upright=1.4|An animated example of a Brownian motion-like [[random walk]] on a [[torus]]. In the [[scaling limit]], random walk approaches the Wiener process according to [[Donsker's theorem]].]]

In [[mathematics]], Brownian motion is described by the '''Wiener process''', a continuous-time [[stochastic process]] named in honor of [[Norbert Wiener]]. It is one of the best known [[Lévy process]]es ([[càdlàg]] stochastic processes with [[stationary increments|stationary]] [[independent increments]]) and occurs frequently in pure and applied mathematics, [[economy|economics]] and [[physics]].
[[File:Wiener process 3d.png|thumb|A single realisation of three-dimensional Brownian motion for times {{math|0 ≤ ''t'' ≤ 2}}]]

The Wiener process {{math|''W<sub>t</sub>''}} is characterized by four facts:<ref>{{cite book | last = Bass | first = Richard F. | url = https://www.cambridge.org/core/books/stochastic-processes/055A84B1EB586FE3032C0CA7D49598AC | title = Stochastic Processes | date = 2011 | publisher = Cambridge University Press | isbn = 978-1-107-00800-7 | series = Cambridge Series in Statistical and Probabilistic Mathematics | location = Cambridge | doi = 10.1017/cbo9780511997044}}</ref>
# {{math|1=''W''<sub>0</sub> = 0}}
# {{math|''W<sub>t</sub>''}} is [[almost surely]] continuous
# {{math|''W<sub>t</sub>''}} has independent increments
# <math>W_t-W_s\sim \mathcal{N}(0,t-s)</math> {{nowrap|(for <math>0 \leq s \le t</math>).}}
<math>\mathcal{N}(\mu, \sigma^2)</math> denotes the [[normal distribution]] with [[expected value]] {{mvar|μ}} and [[variance]] {{math|''σ''<sup>2</sup>}}. The condition that it has independent increments means that if <math>0 \leq s_1 < t_1 \leq s_2 < t_2</math> then <math>W_{t_1}-W_{s_1}</math> and <math>W_{t_2}-W_{s_2}</math> are independent random variables. In addition, for some [[Filtration (probability theory)|filtration]] {{nowrap|<math>\mathcal{F}_t</math>,}} <math>W_t</math> is <math>\mathcal{F}_t</math> [[measurable]] for all {{nowrap|<math>t \geq 0</math>.}}

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]] {{nowrap|<math>[W_t, W_t] = t</math>.}}

A third characterisation is that the Wiener process has a spectral representation as a sine series whose coefficients are independent <math>\mathcal{N}(0, 1)</math> random variables. This representation can be obtained using the [[Kosambi–Karhunen–Loève theorem]].

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. Unlike the random walk, it is [[scale invariance|scale invariant]].
A d-dimensional [[Gaussian free field]] has been described as "a d-dimensional-time analog of Brownian motion."<ref>{{Cite journal |last=Sheffield |first=Scott |date=9 May 2007 |title=Gaussian free fields for mathematicians |url=https://link.springer.com/10.1007/s00440-006-0050-1 |journal=Probability Theory and Related Fields |language=en |volume=139 |issue=3–4 |pages=521–541 |doi=10.1007/s00440-006-0050-1 |issn=0178-8051}}</ref>

===Statistics===

The Brownian motion can be modeled by a [[random walk]].<ref>{{cite book | last = Weiss | first = G. H. | year = 1994 | title = Aspects and applications of the random walk | publisher = North Holland }}</ref>

In the general case, Brownian motion is a [[Markov process]] and described by [[Stochastic calculus|stochastic integral equations]].<ref name="Morozov-2011">{{cite journal | last1 = Morozov | first1 = A. N. | last2 = Skripkin | first2 = A. V. | year = 2011 | title = Spherical particle Brownian motion in viscous medium as non-Markovian random process | journal = [[Physics Letters A]] | volume = 375 | issue = 46 | pages = 4113–4115 | bibcode = 2011PhLA..375.4113M | doi=10.1016/j.physleta.2011.10.001 }}</ref>

===Lévy characterisation===

The French mathematician [[Paul Lévy (mathematician)|Paul Lévy]] proved the following theorem, which gives a necessary and sufficient condition for a continuous {{math|'''R'''<sup>''n''</sup>}}-valued stochastic process {{math|''X''}} to actually be {{mvar|n}}-dimensional Brownian motion. Hence, Lévy's condition can actually be used as an alternative definition of Brownian motion.

Let {{math|1=''X'' = (''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>)}} be a continuous stochastic process on a [[probability space]] {{math|(Ω, Σ, '''P''')}} taking values in {{math|'''R'''<sup>''n''</sup>}}. Then the following are equivalent:

# {{math|''X''}} is a Brownian motion with respect to {{math|'''P'''}}, i.e., the law of {{math|''X''}} with respect to {{math|'''P'''}} is the same as the law of an {{mvar|n}}-dimensional Brownian motion, i.e., the [[push-forward measure]] {{math|''X''<sub>∗</sub>('''P''')}} is [[classical Wiener measure]] on {{math|''C''<sub>0</sub>({{closed-open|0, ∞}}; '''R'''<sup>''n''</sup>)}}.
# both
## {{math|''X''}} is a [[martingale (probability theory)|martingale]] with respect to {{math|'''P'''}} (and its own [[natural filtration]]); and
## for all {{math|1 ≤ ''i'', ''j'' ≤ ''n''}}, {{math|''X''<sub>''i''</sub>(''t'') ''X''<sub>''j''</sub>(''t'') − ''δ''<sub>''ij''</sub> ''t''}} is a martingale with respect to {{math|'''P'''}} (and its own [[natural filtration]]), where {{math|''δ''<sub>''ij''</sub>}} denotes the [[Kronecker delta]].

===Spectral content===

The spectral content of a stochastic process <math>X_t</math> can be found from the [[power spectral density]], formally defined as
<math display="block">S(\omega)=\lim_{T\to\infty}\frac{1}{T}\mathbb{E}\left\{ \left|\int^T_0 e^{i \omega t} X_t dt \right|^2\right\}, </math>
where <math>\mathbb{E}</math> stands for the [[expected value]]. The power spectral density of Brownian motion is found to be<ref>{{cite book | title = Fundamentals of Noise and Vibration Analysis for Engineers by M. P. Norton | last1 = Karczub | first1 = D. G.| last2 = Norton | first2 = M. P.| date = 2003 | language = en | doi = 10.1017/cbo9781139163927 | isbn = 9781139163927}}</ref>
<math display="block">S_{BM}(\omega) = \frac{4 D}{\omega^2}.</math>
where {{mvar|D}} is the [[diffusion coefficient]] of {{math|''X<sub>t</sub>''}}. For naturally occurring signals, the spectral content can be found from the power spectral density of a single realization, with finite available time, i.e.,
<math display="block">S^{(1)}(\omega,T) = \frac{1}{T}\left|\int^T_0 e^{i \omega t}X_t dt\right|^2 ,</math>
which for an individual realization of a Brownian motion trajectory,<ref name="Krapf-2018">{{cite journal | last1 = Krapf | first1 = Diego | last2 = Marinari | first2 = Enzo | last3 = Metzler | first3 = Ralf | last4 = Oshanin | first4 = Gleb | last5 = Xu | first5 = Xinran | last6 = Squarcini | first6 = Alessio | date = 2018 | title = Power spectral density of a single Brownian trajectory: what one can and cannot learn from it | url = https://iopscience.iop.org/article/10.1088/1367-2630/aaa67c | journal = New Journal of Physics | language = en | volume = 20 | issue = 2 | pages = 023029 | doi = 10.1088/1367-2630/aaa67c | issn = 1367-2630 | arxiv = 1801.02986 | bibcode = 2018NJPh...20b3029K | s2cid = 485685 }}</ref> it is found to have expected value <math>\mu_{BM}(\omega,T)</math>
<math display="block">\mu_\text{BM}(\omega,T) = \frac{4 D}{\omega^2} \left[1 - \frac{\sin\left(\omega T\right)}{\omega T}\right]</math>
and [[variance]] <math>\sigma_\text{BM}^2(\omega,T)</math><ref name="Krapf-2018" />
<math display="block">\sigma_S^2(f,T)
= \mathbb{E}\left\{\left(S^{(j)}_T(f)\right)^2\right\}-\mu_S^2(f,T)
=\frac{20 D^2}{f^4}\left[
1-\Big(6-\cos\left(f T\right)\Big) \frac{2\sin\left(f T\right)}{5fT}
+\frac{\Big(17-\cos\left(2fT\right) - 16\cos\left(f T\right)\Big)}{10 f^2 T^2}
\right].</math>

For sufficiently long realization times, the expected value of the power spectrum of a single trajectory converges to the formally defined power spectral density {{nowrap|<math>S(\omega)</math>,}} but its coefficient of variation <math>\gamma = \sigma / \mu</math> tends to {{nowrap|<math>\sqrt{5}/2</math>.}} This implies the distribution of <math>S^{(1)}(\omega,T)</math> is broad even in the infinite time limit.

===Riemannian manifolds===
[[File:BMonSphere.jpg|thumb|upright=1.8|Brownian motion on a sphere]]
Brownian motion is usually considered to take place in [[Euclidean space]]. It is natural to consider how such motion generalizes to more complex shapes, such as [[surface]]s or higher dimensional [[manifold]]s. The formalization requires the space to possess some form of a [[derivative]], as well as a [[metric space|metric]], so that a [[Laplacian]] can be defined. Both of these are available on [[Riemannian manifold]]s.

Riemannian manifolds have the property that [[geodesic]]s can be described in [[polar coordinates]]; that is, displacements are always in a radial direction, at some given angle. Uniform random motion is then described by Gaussians along the radial direction, independent of the angle, the same as in Euclidean space.

The [[Infinitesimal generator (stochastic processes)|infinitesimal generator]] (and hence [[characteristic operator]]) of Brownian motion on Euclidean {{math|'''R'''<sup>''n''</sup>}} is {{math|{{sfrac|1|2}}Δ}}, where {{math|Δ}} denotes the [[Laplace operator]]. Brownian motion on an {{mvar|m}}-dimensional [[Riemannian manifold]] {{math|(''M'', ''g'')}} can be defined as diffusion on {{mvar|M}} with the characteristic operator given by {{math|{{sfrac|1|2}}Δ<sub>LB</sub>}}, half the [[Laplace–Beltrami operator]] {{math|Δ<sub>LB</sub>}}.

One of the topics of study is a characterization of the [[Poincaré recurrence theorem|Poincaré recurrence time]] for such systems.<ref name=Grigoryan-1999/>