{{Short description|Space of stochastic processes}} {{More citations needed|date=February 2023}} thumb|upright|Norbert Wiener

In mathematics, '''classical Wiener space''' is the collection of all continuous functions on a given domain (usually a subinterval of the real line), taking values in a metric space (usually ''n''-dimensional Euclidean space). Classical Wiener space is useful in the study of stochastic processes whose sample paths are continuous functions. It is named after the American mathematician Norbert Wiener.

==Definition==

Consider <math>E\subseteq \mathbb{R}^n</math> and a metric space <math>(M,d)</math>. The '''classical Wiener space''' <math>C(E,M)</math> is the space of all continuous functions <math>f:E\to M.</math> That is, for every fixed <math>t\in E,</math>

:<math>d(f(s), f(t)) \to 0</math> as <math>| s - t | \to 0.</math>

In almost all applications, one takes <math>E=[0,T]</math> or <math>E=\R_+=[0, +\infty)</math> and <math>M=\mathbb{R}^n</math> for some <math>n\in\mathbb{N}.</math> For brevity, write <math>C</math> for <math>C([0,T]);</math> this is a vector space. Write <math>C_0</math> for the linear subspace consisting only of those functions that take the value zero at the infimum of the set <math>E.</math> Many authors refer to <math>C_0</math> as "classical Wiener space".

==Properties of classical Wiener space==

===Uniform topology===

The vector space <math>C</math> can be equipped with the uniform norm

:<math>\| f \| := \sup_{t \in [0,\,T]} |f(t)|</math>

turning it into a normed vector space (in fact a Banach space since <math>[0,T]</math> is compact). This norm induces a metric on <math>C</math> in the usual way: <math>d (f, g) := \| f-g \|</math>. The topology generated by the open sets in this metric is the topology of uniform convergence on <math>[0,T],</math> or the uniform topology.

Thinking of the domain <math>[0,T]</math> as "time" and the range <math>\R^n</math> as "space", an intuitive view of the uniform topology is that two functions are "close" if we can "wiggle space slightly" and get the graph of <math>f</math> to lie on top of the graph of <math>g</math>, while leaving time fixed. Contrast this with the Skorokhod topology, which allows us to "wiggle" both space and time.

If one looks at the more general domain <math>\R_{+}</math> with :<math>\| f \| := \sup_{t \geq 0} |f(t)|,</math> then the Wiener space is no longer a Banach space, however it can be made into one if the Wiener space is defined under the additional constraint :<math>\lim\limits_{s\to\infty}s^{-1}|f(s)|=0.</math>

===Separability and completeness===

With respect to the uniform metric, <math>C</math> is both a separable and a complete space: * Separability is a consequence of the Stone–Weierstrass theorem; * Completeness is a consequence of the fact that the uniform limit of a sequence of continuous functions is itself continuous.

Since it is both separable and complete, <math>C</math> is a Polish space.

===Tightness in classical Wiener space===

Recall that the modulus of continuity for a function <math>f:[0,T]\to\R^n</math> is defined by

:<math>\omega_{f} (\delta) := \sup \left\{ |f(s) - f(t)| : s, t \in [0, T],\, |s - t| \leq \delta \right\}.</math>

This definition makes sense even if <math>f</math> is not continuous, and it can be shown that <math>f</math> is continuous if and only if its modulus of continuity tends to zero as <math>\delta\to 0:</math>

:<math>f \in C \iff \omega_{f} (\delta) \to 0 \text{ as } \delta \to 0</math>.

By an application of the Arzelà-Ascoli theorem, one can show that a sequence <math>(\mu_{n})_{n = 1}^{\infty}</math> of probability measures on classical Wiener space <math>C</math> is tight if and only if both the following conditions are met:

:<math>\lim_{a \to \infty} \limsup_{n \to \infty} \mu_{n} \{ f \in C : | f(0) | \geq a \} = 0,</math> and :<math>\lim_{\delta \to 0} \limsup_{n \to \infty} \mu_{n} \{ f \in C : \omega_{f} (\delta) \geq \varepsilon \} = 0</math> for all <math>\varepsilon >0.</math>

===Classical Wiener measure===

There is a "standard" measure on <math>C_0,</math> known as '''classical Wiener measure''' (or simply '''Wiener measure'''). Wiener measure has (at least) two equivalent characterizations:

If one defines Brownian motion to be a Markov stochastic process <math>B:[0,T]\times\Omega\to\R^n,</math> starting at the origin, with almost surely continuous paths and independent increments

:<math>B_{t} - B_{s} \sim\, \mathrm{Normal} \left( 0, |t - s| \right),</math>

then classical Wiener measure <math>\gamma</math> is the law of the process <math>B.</math>

Alternatively, one may use the abstract Wiener space construction, in which classical Wiener measure <math>\gamma</math> is the radonification of the canonical Gaussian cylinder set measure on the Cameron-Martin Hilbert space corresponding to <math>C_0.</math>

Classical Wiener measure is a Gaussian measure: in particular, it is a strictly positive probability measure.

Given classical Wiener measure <math>\gamma</math> on <math>C_0,</math> the product measure <math>\gamma^n\times\gamma</math> is a probability measure on <math>C</math>, where <math>\gamma^n</math> denotes the standard Gaussian measure on <math>\R^n.</math>

=== Coordinate maps for the Wiener measure === For a stochastic process <math>\{X_t,t\in [0,T]\}:(\Omega,\mathcal{F},P)\to (M,\mathcal{B})</math> and the function space <math>M^E\equiv\{E\to M\}</math> of all functions from <math>E</math> to <math>M</math>, one looks at the map <math>\varphi:\Omega\to M^E</math>. One can then define the ''coordinate maps'' or ''canonical versions'' <math>Y_t:M^E\to M</math> defined by <math>Y_t(\omega)=\omega(t)</math>. The <math>\{Y_t,t\in E\}</math> form another process. For <math>M=\mathbb{R}</math> and <math>E=\R_{+}</math>, the ''Wiener measure'' is then the unique measure on <math>C_0(\R_{+},\R)</math> such that the coordinate process is a Brownian motion.<ref>{{cite book|title=Continuous Martingales and Brownian Motion|first1=Daniel|last1=Revuz|first2=Marc|last2=Yor|date=1999|series=Grundlehren der mathematischen Wissenschaften|volume=293|publisher=Springer|pages=33–37}}</ref>

=== Subspaces of the Wiener space === Let <math>H\subset C_0([0,R])</math> be a Hilbert space that is continuously embbeded and let <math>\gamma</math> be the Wiener measure then <math>\gamma(H)=0</math>. This was proven in 1973 by Smolyanov and Uglanov and in the same year independently by Guerquin.<ref>{{cite journal |first1=Oleg G. |last1=Smolyanov |first2=Alexei V. |last2=Uglanov |title=Every Hilbert subspace of a Wiener space has measure zero |journal=Mathematical Notes |volume=14 |number=3 | date=1973| pages=772–774 |doi=10.1007/BF01147453}}</ref><ref>{{cite journal|first=Małgorzata |last=Guerquin |title=Non-hilbertian structure of the Wiener measure |journal=Colloq. Math. |volume=28 |pages=145–146 |date=1973|doi=10.4064/cm-28-1-145-146 }}</ref> However, there exists a Hilbert space <math>H\subset C_0([0,R])</math> with weaker topology such that <math>\gamma(H)=1</math> which was proven in 1993 by Uglanov.<ref>{{cite journal|first1=Alexei V. |last1=Uglanov |title=Hilbert supports of Wiener measure |journal=Math Notes |volume=51 |number=6 |date=1992 |pages=589–592 |doi=10.1007/BF01263304}}</ref>

==See also==

* Abstract Wiener space * Gaussian probability space * Malliavin calculus * Malliavin derivative * Skorokhod space, a generalization of classical Wiener space, which allows functions to be discontinuous * Wiener process

== References == <references /> {{Measure theory}} {{Analysis in topological vector spaces}}

Category:Measure theory Category:Metric geometry Category:Stochastic processes