Abstract Wiener space

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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.

1 Definition
2 Properties
3 Example: Classical Wiener space
4 See also

[edit] 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.

[edit] 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.

[edit] 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

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

with inner product

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