Abstract Wiener space
Encyclopedia
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
-dimension
al 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
.
. 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 γH 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.
, and is known as classical Wiener space. This is the abstract Wiener space with
with inner product
E = C0([0, T]; Rn) with norm
and i : H → E the inclusion map
. The measure γ is called classical Wiener measure or simply Wiener measure.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
object in measure theory, used to construct a "decent" (strictly positive
Strictly positive measure
In mathematics, strict positivity is a concept in measure theory. Intuitively, a strictly positive measure is one that is "nowhere zero", or that it is zero "only on points".-Definition:...
and locally finite
Locally finite measure
In mathematics, a locally finite measure is a measure for which every point of the measure space has a neighbourhood of finite measure.-Definition:...
) measure on an infinite
Infinity
Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...
-dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
al vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
. It is named after the American
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Norbert Wiener
Norbert Wiener
Norbert Wiener was an American mathematician.A famous child prodigy, Wiener later became an early researcher in stochastic and noise processes, contributing work relevant to electronic engineering, electronic communication, and control systems.Wiener is regarded as the originator of cybernetics, a...
. Wiener's original construction only applied to the space of real-valued continuous paths
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
on the unit interval
Unit interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1...
, known as classical Wiener space
Classical Wiener space
In mathematics, classical Wiener space is the collection of all continuous functions on a given domain , taking values in a metric space . Classical Wiener space is useful in the study of stochastic processes whose sample paths are continuous functions...
; Leonard Gross provided the generalization to the case of a general separable Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
.
Definition
Let H be a separable Hilbert spaceHilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
. Let E be a separable Banach space. Let i : H → E be an injective continuous linear map with dense
Dense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...
image
Image (mathematics)
In mathematics, an image is the subset of a function's codomain which is the output of the function on a subset of its domain. Precisely, evaluating the function at each element of a subset X of the domain produces a set called the image of X under or through the function...
(i.e., the closure
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
of i(H) in E is E itself) that radonifies the canonical Gaussian cylinder set measure γH 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
Identity function
In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument...
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-algebraBorel algebraIn mathematics, a Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement...
generated by the open subsetsOpen setThe concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
of E. - γ is a Gaussian measureGaussian measureIn mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space Rn, closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces...
in the sense that f∗(γ) is a Gaussian measure on R for every linear functionalLinear functionalIn linear algebra, a linear functional or linear form is a linear map from a vector space to its field of scalars. In Rn, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the...
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 isomorphicIsomorphismIn abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
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 translationTranslation (geometry)In Euclidean geometry, a translation moves every point a constant distance in a specified direction. A translation can be described as a rigid motion, other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant vector to every point, or...
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.e., the abstract Wiener measure γ12 on the Cartesian productCartesian productIn mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
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 lawKolmogorov's zero-one lawIn probability theory, Kolmogorov's zero-one law, named in honor of Andrey Nikolaevich Kolmogorov, specifies that a certain type of event, called a tail event, will either almost surely happen or almost surely not happen; that is, the probability of such an event occurring is zero or one.Tail...
.
Example: Classical Wiener space
Arguably the most frequently-used abstract Wiener space is the space of continuous pathsPath (topology)
In mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to XThe initial point of the path is f and the terminal point is f. One often speaks of a "path from x to y" where x and y are the initial and terminal points of the path...
, and is known as classical Wiener space. This is the abstract Wiener space with
with inner product
E = C0([0, T]; Rn) with norm
Norm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
and i : H → E the inclusion map
Inclusion map
In mathematics, if A is a subset of B, then the inclusion map is the function i that sends each element, x of A to x, treated as an element of B:i: A\rightarrow B, \qquad i=x....
. The measure γ is called classical Wiener measure or simply Wiener measure.
See also
- Structure theorem for Gaussian measuresStructure theorem for Gaussian measuresIn mathematics, the structure theorem for Gaussian measures shows that the abstract Wiener space construction is essentially the only way to obtain a strictly positive Gaussian measure on a separable Banach space...
- There is no infinite-dimensional Lebesgue measureThere is no infinite-dimensional Lebesgue measureIn mathematics, it is a theorem that there is no analogue of Lebesgue measure on an infinite-dimensional Banach space. Other kinds of measures are therefore used on infinite-dimensional spaces: often, the abstract Wiener space construction is used...