Classical Wiener space

Properties of classical Wiener space

Uniform topology

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

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

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: ${\displaystyle d(f,g):=\Vert f-g\Vert }$. 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

${\displaystyle \Vert f\Vert :=\underset{t\ge 0}{sup}|f(t)|,}$

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

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

Separability and completeness

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.

Tightness in classical Wiener space

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

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

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 ${\displaystyle \delta \to 0:}$

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

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

${\displaystyle \underset{a\to \mathrm{\infty }}{lim}\underset{n\to \mathrm{\infty }}{lim sup}{\mu }_n\{f\in C:|f(0)|\ge a\}=0,}$ and

${\displaystyle \underset{\delta \to 0}{lim}\underset{n\to \mathrm{\infty }}{lim sup}{\mu }_n\{f\in C:{\omega }_f(\delta )\ge \epsilon \}=0}$ for all ${\displaystyle \epsilon >0.}$

Classical Wiener measure

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 ${\displaystyle B:[0,T]\times \mathrm{\Omega }\to {\mathbb{R}}^n,}$ starting at the origin, with almost surely continuous paths and independent increments

${\displaystyle B_t-B_s\sim \mathrm{N}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}\left(0,|t-s|\right),}$

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 ${\displaystyle {\mathbb{R}}^n.}$

Coordinate maps for the Wiener measure

For a stochastic process ${\displaystyle \{X_t,t\in [0,T]\}:(\mathrm{\Omega },\mathcal{F},P)\to (M,\mathcal{B})}$ 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 ${\displaystyle \phi :\mathrm{\Omega }\to M^E}$. 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.

Subspaces of the Wiener space

Let ${\displaystyle H\subset C_0([0,R])}$ be a Hilbert space that is continuously embedded 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. 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.