Classical Wiener space
# Classical Wiener space

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 E ⊆ $R^n$ and a metric space (M, d). The classical Wiener space C(E; M) is the space of all continuous functions f : E → 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, +∞) and M = $R^n$ for some n in N. For brevity, write C for C([0, T ]; $R^n);$ this is a vector space. Write $C_0$ for the linear subspace consisting only of those functions that take the value zero at the infimum of the set E. Many authors refer to $C_0$ as "classical Wiener space".

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

## Properties of classical Wiener space

### Uniform topology

The vector space C can be equipped with the uniform norm

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

turning it into a normed vector space (in fact a Banach space). 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 $R^n$ 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 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.

### 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.

### Tightness in classical Wiener space

Recall that the modulus of continuity for a function f : [0, T ] → $R^n$ is defined by

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

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

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

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

    $\lim _{a\to \infty }\limsup _{n\to \infty }\mu _{n}\{f\in C\mid |f(0)|\geq a\}=0,$ and
    $\lim _{\delta \to 0}\limsup _{n\to \infty }\mu _{n}\{f\in C\mid \omega _{f}(\delta )\geq \varepsilon \}=0$ for all ε > 0.

### 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 ] × Ω → $R^n,$ starting at the origin, with almost surely continuous paths and independent 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 $γ^n$ × γ is a probability measure on C, where $γ^n$ denotes the standard Gaussian measure on $R^n.$

