# Classical Wiener space

In mathematics, **classical Wiener space** is 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.

## 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*,

- as

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".

## Properties of classical Wiener space

### Uniform topology

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

turning it into a normed vector space (in fact a Banach space). This norm induces a metric on *C* in the usual way: . 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 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.

### 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

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:

- as δ → 0.

By an application of the Arzelà-Ascoli theorem, one can show that a sequence of probability measures on classical Wiener space *C* is tight if and only if both the following conditions are met:

- and
- 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

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}.

## See also

- Skorokhod space, a generalization of classical Wiener space, which allows functions to be discontinuous
- Abstract Wiener space
- Wiener process