Classical Wiener space
Encyclopedia
In mathematics
, classical Wiener space is the collection of all continuous function
s on a given domain (usually a sub-interval
of the real line
), taking values in a metric space
(usually n-dimensional Euclidean space
). Classical Wiener space is useful in the study of stochastic processes whose sample paths are continuous functions. It is named after the American
mathematician
Norbert Wiener
.
and a metric space 
, the classical Wiener space 
is the space of all continuous functions 
: i.e., for every (fixed) 
,
as 
In almost all applications, one takes
or 
and 
for some 
. For brevity, write 
for 
; this is a vector space
. Write
for the linear subspace
consisting only of those paths that start at the origin. (Many authors refer to
as "classical Wiener space".)
The more general case is treated in, for example, Billingsley.
can be equipped with the uniform norm
turning it into a normed vector space
(and indeed into a Banach space
as we will see below.) This norm induces a metric
on
in the usual way: 
. The topology
generated by the open set
s in this metric is the topology of uniform convergence on
, or the uniform topology
.
Thinking of the domain
as "time" and the range 
as "space", an intuitive view of the uniform topology is that two functions are "close" if we can "wiggle space a bit" and get the graph of 
to lie on top of the graph of 
, while leaving time fixed. Contrast this with the Skorokhod topology, which allows us to "wiggle" both space and time.
is both a separable and a complete space
:
Since it is both separable and complete,
is a Polish space
.
for a function
is defined by
This definition makes sense even if
is not continuous, and it can be shown that 
is continuous if and only if
its modulus of continuity tends to zero as
:
as 
By an application of the Arzelà-Ascoli theorem
, one can show that a sequence
of probability measure
s on classical Wiener space
is tight
if and only if both the following conditions are met:
and
for all 
, known as classical Wiener measure (or simply Wiener measure). Wiener measure has (at least!) two equivalent characterizations:
If one defines Brownian motion
to be a Markov
stochastic process
, starting at the origin, with almost surely
continuous paths and independent increments
then classical Wiener measure
is the law
of the process
.
Alternatively, one may use the abstract Wiener space
construction, in which classical Wiener measure
is the radonification
of the canonical Gaussian cylinder set measure on the Cameron-Martin Hilbert space
corresponding to
.
Classical Wiener measure is a Gaussian measure
: in particular, it is a strictly positive
probability measure.
Given classical Wiener measure
on 
, the product measure
is a probability measure on 
, where 
denotes the standard Gaussian measure
on
.
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...
, classical Wiener space is the collection of all continuous function
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...
s on a given domain (usually a sub-interval
Interval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
of the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
), taking values in a metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
(usually n-dimensional Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
). Classical Wiener space is useful in the study of stochastic processes whose sample paths are continuous functions. 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...
.
Definition
Given and a metric space , the classical Wiener space is the space of all continuous functions : i.e., for every (fixed) , as In almost all applications, one takes
or and for some . For brevity, write for ; this is a vector spaceVector 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...
. Write
for the linear subspaceLinear subspace
The concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....
consisting only of those paths that start at the origin. (Many authors refer to
as "classical Wiener space".)The more general case is treated in, for example, Billingsley.
Uniform topology
The vector space can be equipped with the uniform normturning it into a normed vector space
Normed vector space
In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rn. The following properties of "vector length" are crucial....
(and indeed into a 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...
as we will see below.) This norm induces a metric
Metric (mathematics)
In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...
on
in the usual way: . The topologyTopology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
generated by the open set
Open set
The 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...
s in this metric is the topology of uniform convergence on
, or the uniform topologyUniform topology
In mathematics, the uniform topology on a space has several different meanings depending on the context:* In functional analysis, it sometimes refers to a polar topology on a topological vector space....
.
Thinking of the domain
as "time" and the range as "space", an intuitive view of the uniform topology is that two functions are "close" if we can "wiggle space a bit" and get the graph of to lie on top of the graph of , while leaving time fixed. Contrast this with the Skorokhod topology, which allows us to "wiggle" both space and time.Separability and completeness
With respect to the uniform metric, is both a separable and a complete spaceComplete space
In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....
:
- separability is a consequence of the Stone-Weierstrass theoremStone-Weierstrass theoremIn mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on an interval [a,b] can be uniformly approximated as closely as desired by a polynomial function...
; - completeness is a consequence of the fact that the uniform limit of a sequence of continuous functions is itself continuous.
Since it is both separable and complete,
is a Polish spacePolish space
In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish...
.
Tightness in classical Wiener space
Recall that the modulus of continuityModulus of continuity
In mathematical analysis, a modulus of continuity is a function\omega:[0,\infty]\to[0,\infty]used to measure quantitatively the uniform continuity of functions. So, a function f:I\to\R admits \omega as a modulus of continuity if and only if|f-f|\leq\omega,for all x and y in the domain of f...
for a function
is defined byThis definition makes sense even if
is not continuous, and it can be shown that is continuous if and only ifIf and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
its modulus of continuity tends to zero as
: as By an application of the Arzelà-Ascoli theorem
Arzelà-Ascoli theorem
In mathematics, the Arzelà–Ascoli theorem of functional analysis gives necessary and sufficient conditions to decide whether every subsequence of a given sequence of real-valued continuous functions defined on a closed and bounded interval has a uniformly convergent subsequence. The main condition...
, one can show that a sequence
of probability measureProbability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity...
s on classical Wiener space
is tightTightness of measures
In mathematics, tightness is a concept in measure theory. The intuitive idea is that a given collection of measures does not "escape to infinity."-Definitions:...
if and only if both the following conditions are met:
and for all Classical Wiener measure
There is a "standard" measure on, known as classical Wiener measure (or simply Wiener measure). Wiener measure has (at least!) two equivalent characterizations:If one defines Brownian motion
Brownian motion
Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...
to be a Markov
Markov property
In probability theory and statistics, the term Markov property refers to the memoryless property of a stochastic process. It was named after the Russian mathematician Andrey Markov....
stochastic process
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
, starting at the origin, with almost surelyAlmost surely
In probability theory, one says that an event happens almost surely if it happens with probability one. The concept is analogous to the concept of "almost everywhere" in measure theory...
continuous paths and independent increments
then classical Wiener measure
is the lawLaw (stochastic processes)
In mathematics, the law of a stochastic process is the measure that the process induces on the collection of functions from the index set into the state space...
of the process
.Alternatively, one may use the abstract Wiener space
Abstract Wiener space
An abstract Wiener space is a mathematical object in measure theory, used to construct a "decent" measure on an infinite-dimensional vector space. It is named after the American mathematician Norbert Wiener...
construction, in which classical Wiener measure
is the radonificationRadonifying function
In measure theory, a radonifying function between measurable spaces is one that takes a cylinder set measure on the first space to a true measure on the second space...
of the canonical Gaussian cylinder set measure on the Cameron-Martin Hilbert space
Hilbert 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...
corresponding to
.Classical Wiener measure is a Gaussian measure
Gaussian measure
In 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 particular, it is a 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:...
probability measure.
Given classical Wiener measure
on , the product measureProduct measure
In mathematics, given two measurable spaces and measures on them, one can obtain the product measurable space and the product measure on that space...
is a probability measure on , where denotes the standard Gaussian measureGaussian measure
In 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...
on
.See also
- Skorokhod spaceCàdlàgIn mathematics, a càdlàg , RCLL , or corlol function is a function defined on the real numbers that is everywhere right-continuous and has left limits everywhere...
, a generalization of classical Wiener space, which allows functions to be discontinuous - Abstract Wiener spaceAbstract Wiener spaceAn abstract Wiener space is a mathematical object in measure theory, used to construct a "decent" measure on an infinite-dimensional vector space. It is named after the American mathematician Norbert Wiener...
- Wiener processWiener processIn mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown...