Classical Wiener space

In mathematics, classical Wiener space is the collection of all continuous functions on a given domain, taking values in a metric 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 and a metric space. The classical Wiener space is the space of all continuous functions That is, for every fixed
In almost all applications, one takes or and for some For brevity, write for this is a vector space. Write for the linear subspace consisting only of those functions that take the value zero at the infimum of the set Many authors refer to as "classical Wiener space".

Properties of classical Wiener space

Uniform topology

The vector space can be equipped with the uniform norm
turning it into a normed vector space. This norm induces a metric on in the usual way:. The topology generated by the open sets in this metric is the topology of uniform convergence on or the uniform topology.
Thinking of the domain as "time" and the range 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 to lie on top of the graph of, 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 with
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

Separability and completeness

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

Tightness in classical Wiener space

Recall that the modulus of continuity for a function is defined by
This definition makes sense even if is not continuous, and it can be shown that is continuous if and only if its modulus of continuity tends to zero as
By an application of the Arzelà-Ascoli theorem, one can show that a sequence of probability measures on classical Wiener space is tight if and only if both the following conditions are met:

Classical Wiener measure

There is a "standard" measure on known as classical Wiener measure. Wiener measure has two equivalent characterizations:
If one defines Brownian motion to be a Markov stochastic process starting at the origin, with almost surely continuous paths and independent increments
then classical Wiener measure is the law of the process
Alternatively, one may use the abstract Wiener space construction, in which classical Wiener measure is the radonification of the canonical Gaussian cylinder set measure on the Cameron-Martin Hilbert space corresponding to
Classical Wiener measure is a Gaussian measure: in particular, it is a strictly positive probability measure.
Given classical Wiener measure on the product measure is a probability measure on, where denotes the standard Gaussian measure on

Coordinate maps for the Wiener measure

For a stochastic process and the function space of all functions from to, one looks at the map. One can then define the coordinate maps or canonical versions defined by. The form another process. For and, the Wiener measure is then the unique measure on such that the coordinate process is a Brownian motion.

Subspaces of the Wiener space

Let be a Hilbert space that is continuously embbeded and let be the Wiener measure then. This was proven in 1973 by Smolyanov and Uglanov and in the same year independently by Guerquin. However, there exists a Hilbert space with weaker topology such that which was proven in 1993 by Uglanov.