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.

DefinitionSome 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.

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 Rⁿ for some n ∈ N. Setting H = Rⁿ and i : H → E to be the canonical isomorphism gives the abstract Wiener measure γ = γⁿ, the standard Gaussian measure on Rⁿ.
The behaviour of γ under translation is described by the Cameron–Martin theorem.
Given two abstract Wiener spaces i₁ : H₁ → E₁ and i₂ : H₂ → E₂, one can show that γ₁₂ = γ₁ ⊗ γ₂. In full:

\( (i_{1} \times i_{2})_{*} (\gamma^{H_{1} \times H_{2}}) = (i_{1})_{*} \left( \gamma^{H_{1}} \right) \otimes (i_{2})_{*} \left( \gamma^{H_{2}} \right), \)

i.e., the abstract Wiener measure γ₁₂ on the Cartesian product E₁ × E₂ is the product of the abstract Wiener measures on the two factors E₁ and E₂.

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
Main article: 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

\( H := L_{0}^{2, 1} ([0, T]; \mathbb{R}^{n}) := \{ \text{paths starting at 0 with first derivative} \in L^{2} \}, \)

with inner product

\( \langle \sigma_{1}, \sigma_{2} \rangle_{L_{0}^{2,1}} := \int_{0}^{T} \langle \dot{\sigma}_{1} (t), \dot{\sigma}_{2} (t) \rangle_{\mathbb{R}^{n}} \, \mathrm{d} t, , \)

E = C0([0, T]; Rn) with norm

\( \| \sigma \|_{C_{0}} := \sup_{t \in [0, T]} \| \sigma (t) \|_{\mathbb{R}^{n}},, \)

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