Classical Wiener space

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Classical Wiener space" – news · newspapers · books · scholar · JSTOR (February 2023) (Learn how and when to remove this message)

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.

Consider ${\displaystyle E\subseteq \mathbb {R} ^{n}}$ and a metric space ${\displaystyle (M,d)}$. The classical Wiener space ${\displaystyle C(E,M)}$ is the space of all continuous functions ${\displaystyle f:E\to M.}$ That is, for every fixed ${\displaystyle t\in E,}$

${\displaystyle d(f(s),f(t))\to 0}$ as ${\displaystyle |s-t|\to 0.}$

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

The vector space ${\displaystyle C}$ can be equipped with the uniform norm

${\displaystyle \|f\|:=\sup _{t\in [0,\,T]}|f(t)|}$

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

Thinking of the domain ${\displaystyle [0,T]}$ as "time" and the range ${\displaystyle \mathbb {R} ^{n}}$ 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 ${\displaystyle f}$ to lie on top of the graph of ${\displaystyle g}$, 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 ${\displaystyle \mathbb {R} _{+}}$ with

${\displaystyle \|f\|:=\sup _{t\geq 0}|f(t)|,}$

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

${\displaystyle \lim \limits _{s\to \infty }s^{-1}|f(s)|=0.}$

With respect to the uniform metric, ${\displaystyle C}$ 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, ${\displaystyle C}$ is a Polish space.

Recall that the modulus of continuity for a function ${\displaystyle f:[0,T]\to \mathbb {R} ^{n}}$ is defined by

${\displaystyle \omega _{f}(\delta ):=\sup \left\{|f(s)-f(t)|:s,t\in [0,T],\,|s-t|\leq \delta \right\}.}$

This definition makes sense even if ${\displaystyle f}$ is not continuous, and it can be shown that ${\displaystyle f}$ is continuous if and only if its modulus of continuity tends to zero as ${\displaystyle \delta \to 0:}$

${\displaystyle f\in C\iff \omega _{f}(\delta )\to 0{\text{ as }}\delta \to 0}$.

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

${\displaystyle \lim _{a\to \infty }\limsup _{n\to \infty }\mu _{n}\{f\in C:|f(0)|\geq a\}=0,}$ and

${\displaystyle \lim _{\delta \to 0}\limsup _{n\to \infty }\mu _{n}\{f\in C:\omega _{f}(\delta )\geq \varepsilon \}=0}$ for all ${\displaystyle \varepsilon >0.}$

There is a "standard" measure on ${\displaystyle C_{0},}$ 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 ${\displaystyle B:[0,T]\times \Omega \to \mathbb {R} ^{n},}$ starting at the origin, with almost surely continuous paths and independent increments

${\displaystyle B_{t}-B_{s}\sim \,\mathrm {Normal} \left(0,|t-s|\right),}$

then classical Wiener measure ${\displaystyle \gamma }$ is the law of the process ${\displaystyle B.}$

Alternatively, one may use the abstract Wiener space construction, in which classical Wiener measure ${\displaystyle \gamma }$ is the radonification of the canonical Gaussian cylinder set measure on the Cameron-Martin Hilbert space corresponding to ${\displaystyle C_{0}.}$

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

Given classical Wiener measure ${\displaystyle \gamma }$ on ${\displaystyle C_{0},}$ the product measure ${\displaystyle \gamma ^{n}\times \gamma }$ is a probability measure on ${\displaystyle C}$, where ${\displaystyle \gamma ^{n}}$ denotes the standard Gaussian measure on ${\displaystyle \mathbb {R} ^{n}.}$

For a stochastic process ${\displaystyle \{X_{t},t\in [0,T]\}:(\Omega ,{\mathcal {F}},P)\to (M,{\mathcal {B}})}$ and the function space ${\displaystyle M^{E}\equiv \{E\to M\}}$ of all functions from ${\displaystyle E}$ to ${\displaystyle M}$, one looks at the map ${\displaystyle \varphi :\Omega \to M^{E}}$. One can then define the coordinate maps or canonical versions ${\displaystyle Y_{t}:M^{E}\to M}$ defined by ${\displaystyle Y_{t}(\omega )=\omega (t)}$. The ${\displaystyle \{Y_{t},t\in E\}}$ form another process. For ${\displaystyle M=\mathbb {R} }$ and ${\displaystyle E=\mathbb {R} _{+}}$, the Wiener measure is then the unique measure on ${\displaystyle C_{0}(\mathbb {R} _{+},\mathbb {R} )}$ such that the coordinate process is a Brownian motion.^[1]

Let ${\displaystyle H\subset C_{0}([0,R])}$ be a Hilbert space that is continuously embbeded and let ${\displaystyle \gamma }$ be the Wiener measure then ${\displaystyle \gamma (H)=0}$. This was proven in 1973 by Smolyanov and Uglanov and in the same year independently by Guerquin.^[2][3] However, there exists a Hilbert space ${\displaystyle H\subset C_{0}([0,R])}$ with weaker topology such that ${\displaystyle \gamma (H)=1}$ which was proven in 1993 by Uglanov.^[4]

• 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