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.

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:

::$(i\_\{1\}\; imes\; i\_\{2\})\_\{*\}\; (gamma^\{H\_\{1\}\; imes\; H\_\{2)\; =\; (i\_\{1\})\_\{*\}\; left(\; gamma^\{H\_\{1\; ight)\; otimes\; (i\_\{2\})\_\{*\}\; left(\; gamma^\{H\_\{2\; ight),$

: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.

* 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

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

with inner product

:$langle\; sigma\_\{1\},\; sigma\_\{2\}\; angle\_\{L\_\{0\}^\{2,1\; :=\; int\_\{0\}^\{T\}\; langle\; dot\{sigma\}\_\{1\}\; (t),\; dot\{sigma\}\_\{2\}\; (t)\; angle\_\{mathbb\{R\}^\{n\; ,\; mathrm\{d\}\; t,$

"E" = "C"₀( [0, "T"] ; R^"n") 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.

ee also

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

References

* cite book
last = Bell
first = Denis R.
title = The Malliavin calculus
publisher = Dover Publications Inc.
location = Mineola, NY
year = 2006
pages = pp. x+113
isbn = 0-486-44994-7 MathSciNet|id=2250060 (See section 1.1)
* cite book
last = Gross
first = Leonard
chapter = Abstract Wiener spaces
title = Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), Vol. II: Contributions to Probability Theory, Part 1
pages = 31--42
publisher = Univ. California Press
location = Berkeley, Calif.
year = 1967 MathSciNet|id=0212152