# Abstract Wiener space

An **abstract Wiener space** is a mathematical object in measure theory, used to construct a "decent" (strictly positive and locally finite) measure on an infinite-dimensional vector space. It is named after the American mathematician Norbert Wiener. Wiener's original construction only applied to the space of real-valued continuous paths on the unit interval, known as classical Wiener space; provided the generalization to the case of a general separable Banach space.

The structure theorem for Gaussian measures states that **all** Gaussian measures can be represented by the abstract Wiener space construction.

## Definition

Let *H* be a separable Hilbert space. Let *E* be a separable Banach space. Let *i* : *H* → *E* be an injective continuous linear map with dense image (i.e., the closure of *i*(*H*) in *E* is *E* itself) that radonifies the canonical Gaussian cylinder set measure *γ*^{H} on *H*. Then the triple (*i*, *H*, *E*) (or simply *i* : *H* → *E*) is called an **abstract Wiener space**. The measure *γ* induced on *E* is called the **abstract Wiener measure** of *i* : *H* → *E*.

The Hilbert space *H* is sometimes called the **Cameron–Martin space** or **reproducing kernel Hilbert space**.

Some sources (e.g. Bell (2006)) consider *H* to be a densely embedded Hilbert subspace of the Banach space *E*, with *i* simply the inclusion of *H* into *E*. There is no loss of generality in taking this "embedded spaces" viewpoint instead of the "different spaces" viewpoint given above.

## Properties

*γ*is a Borel measure: it is defined on the Borel σ-algebra generated by the open subsets of*E*.*γ*is a Gaussian measure in the sense that*f*_{∗}(*γ*) is a Gaussian measure on**R**for every linear functional*f*∈*E*^{∗},*f*≠ 0.- Hence,
*γ*is strictly positive and locally finite. - If
*E*is a finite-dimensional Banach space, we may take*E*to be isomorphic to**R**^{n}for some*n*∈**N**. Setting*H*=**R**^{n}and*i*:*H*→*E*to be the canonical isomorphism gives the abstract Wiener measure*γ*=*γ*^{n}, the standard Gaussian measure on**R**^{n}. - The behaviour of
*γ*under translation is described by the Cameron–Martin theorem. - Given two abstract Wiener spaces
*i*_{1}:*H*_{1}→*E*_{1}and*i*_{2}:*H*_{2}→*E*_{2}, one can show that*γ*_{12}=*γ*_{1}⊗*γ*_{2}. In full:

- i.e., the abstract Wiener measure
*γ*_{12}on the Cartesian product*E*_{1}×*E*_{2}is the product of the abstract Wiener measures on the two factors*E*_{1}and*E*_{2}.

- If
*H*(and*E*) are infinite dimensional, then the image of*H*has measure zero:*γ*(*i*(*H*)) = 0. This fact is a consequence of Kolmogorov's zero-one law.

## Example: Classical Wiener space

Arguably the most frequently-used abstract Wiener space is the space of continuous paths, and is known as **classical Wiener space**. This is the abstract Wiener space with

with inner product

*E* = *C*_{0}([0, *T*]; **R**^{n}) with norm

and *i* : *H* → *E* the inclusion map. The measure *γ* is called **classical Wiener measure** or simply Wiener measure.

## See also

- Structure theorem for Gaussian measures
- There is no infinite-dimensional Lebesgue measure

## References

- Bell, Denis R. (2006).
*The Malliavin calculus*. Mineola, NY: Dover Publications Inc. p. x+113. ISBN 0-486-44994-7. MR 2250060. (See section 1.1) - Gross, Leonard (1967). "Abstract Wiener spaces".
*Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), Vol. II: Contributions to Probability Theory, Part 1*. Berkeley, Calif.: Univ. California Press. pp. 31–42. MR 0212152.