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 $i : H \to E$ be an injective continuous linear map with dense image (i.e., $\overline{i(H)} = E$ ) that radonifies the canonical Gaussian cylinder set measure $γ H$ on $H$ . Then the triple $(i, H, E)$ (or simply $i : H \to E$ ) is called an abstract Wiener space. The measure $γ$ induced on $E$ is called the abstract Wiener measure of $i : H \to E$ .

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 $\ell_{*} (\gamma)$ is a Gaussian measure on $\mathbb{R}$ for every linear functional $\ell \in E^{*}$ , $\ell \neq 0$ .
Hence, $γ$ is strictly positive and locally finite.
If $E$ is a finite-dimensional Banach space, we may take $E$ to be isomorphic to $\mathbb{R}^{n}$ for some $n \in \mathbb{N}$ . Setting $H = \mathbb{R}^{n}$ and $i : H \to E$ to be the canonical isomorphism gives the abstract Wiener measure $γ = γ n$ , the standard Gaussian measure on $\mathbb{R}^{n}$ .
The behaviour of $γ$ under translation is described by the Cameron-Martin theorem.
Given two abstract Wiener spaces $i_{1} : H_{1} \to E_{1}$ and $i_{2} : H_{2} \to E_{2}$ , one can show that $\gamma_{12} = \gamma_{1} \otimes \gamma_{2}$ . 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 on the Cartesian product $E_{1} \times E_{2}$ is the product of the abstract Wiener measures on the two factors $E 1$ and $E 2$ .

[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

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

with inner product

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