Classical Wiener space

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In mathematics, classical Wiener space is the term given to the collection of all continuous functions on a given domain (usually a sub-interval 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.

1 Definition
2 Properties of classical Wiener space
3 See also
4 References

[edit] Definition

Given $E \subseteq \mathbb{R}$ and a metric space $(M, d)$ , the classical Wiener space $C (E; M)$ is the vector space of all continuous functions $f : E \to M$ : i.e., for every (fixed) $t \in E$ ,

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

In almost all applications, one takes $E = [0, T]$ or $[0, + \infty)$ and $M = \mathbb{R}^{n}$ for some $n \in \mathbb{N}$ . For brevity, write $C$ for $C ([0, T]; \mathbb{R}^{n})$ . Write $C 0$ for the linear subspace consisting only of those paths that start at the origin. (Many authors refer to $C 0$ as "classical Wiener space".)

The more general case is treated in, for example, Billingsley.

[edit] Properties of classical Wiener space

[edit] Uniform topology

The vector space $C$ can be equipped with the uniform norm

$\| f \| := \sup_{t \in [0, T]} | f(t) |.$

This norm induces a metric on $C$ in the usual way: $d (f, g) := \| f - g \|$ . The topology generated by the open sets in this metric is the topology of uniform convergence on $[0, T]$ , or the uniform topology.

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

[edit] Separability and completeness

With respect to the uniform metric, $C$ 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, $C$ is a Polish space.

[edit] Tightness in classical Wiener space

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

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

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

$f \in C \iff \varpi_{f} (\delta) \to 0$ as $\delta \to 0.$

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

$\lim_{a \to \infty} \limsup_{n \to \infty} \mu_{n} \{ f \in C | | f(0) | \geq a \} = 0,$ and

$\lim_{\delta \to 0} \limsup_{n \to \infty} \mu_{n} \{ f \in C | \varpi_{f} (\delta) \geq \varepsilon \} = 0$ for all $\varepsilon > 0$ .

[edit] Classical Wiener measure

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

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

then classical Wiener measure $γ$ is the law of the process $B$ .

Alternatively, one may use the abstract Wiener space construction, in which classical Wiener measure $γ$ is the radonification of the canonical Gaussian cylinder set measure on the Cameron-Martin Hilbert space corresponding to $C 0$ .

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

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