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; Leonard Gross 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
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.