In mathematics, classical Wiener space is the collection of all continuous functions 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. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... Look up domain in Wiktionary, the free dictionary. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... In mathematics, the real line is simply the set of real numbers. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In the mathematics of probability, a stochastic process can be thought of as a random function. ... Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... Norbert Wiener Norbert Wiener (November 26, 1894, Columbia, Missouri – March 18, 1964, Stockholm Sweden) was an American theoretical and applied mathematician. ...

Definition

Given and a metric space (M,d), the classical Wiener space C(E;M) is the vector space of all continuous functions : i.e., for every (fixed) , In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...

as

In almost all applications, one takes E = [0,T] or and for some . For brevity, write C for . Write C₀ for the linear subspace consisting only of those paths that start at the origin. (Many authors refer to C₀ as "classical Wiener space".) The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics. ...

The more general case is treated in, for example, Billingsley.

Properties of classical Wiener space

Uniform topology

The vector space C can be equipped with the uniform norm In mathematical analysis, the uniform norm assigns to real- or complex-valued functions f the nonnegative number This norm is also called the supremum norm or the Chebyshev norm. ...

This norm induces a metric on C in the usual way: . The topology generated by the open sets in this metric is the topology of uniform convergence on [0,T], or the uniform topology. In mathematics a metric or distance function is a function which defines a distance between elements of a set. ... A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In the mathematical field of analysis, uniform convergence is a type of convergence stronger than pointwise convergence. ...

Thinking of the domain [0,T] 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 f to lie on top of the graph of g, while leaving time fixed. Contrast this with the Skorokhod topology, which allows us to "wiggle" both space and time. In mathematics, a càdlàg function is a function defined on the real numbers (or a subset of them) that is everywhere right-continuous and has left limits everywhere. ...

Separability and completeness

With respect to the uniform metric, C is both a separable and a complete space: In topology and related areas of mathematics a topological space is called separable if it contains a countable dense subset; that is, a set with a countable number of elements whose closure is the entire space. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For...

• separability is a consequence of the Stone-Weierstrass theorem;
• 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, C is a Polish space. In 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. ... In mathematics, a Polish space is a separable completely metrisable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. ...

Tightness in classical Wiener space

Recall that the modulus of continuity for a function is defined by In mathematics, the modulus of continuity is a precise way to measure the smoothness of a function. ...

This definition makes sense even if f is not continuous, and it can be shown that a function is continuous if and only if its modulus of continuity tends to zero as : It has been suggested that this article or section be merged into Logical biconditional. ...

as

By an application of the Arzelà-Ascoli theorem, one can show that a sequence of probability measures on classical Wiener space C is tight if and only if both the following conditions are met: In mathematics, the Arzelà-Ascoli theorem of functional analysis is a criterion to decide whether a set of continuous functions from a compact metric space into a metric space is compact in the topology of uniform convergence. ... In mathematics, a probability space is a set S, together with a σ-algebra X on S and a measure P on that σ-algebra such that P(S) = 1. ... In mathematics, tightness is a concept in measure theory, the intuitive idea being that a given collection of measures does not escape to infinity. ...

and

for all .

Classical Wiener measure

There is a "standard" measure on C₀, 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 increments Three different views of Brownian motion, with 32 steps, 256 steps, and 2048 steps denoted by progressively lighter colors. ... In probability theory, a stochastic process has the Markov property if the conditional probability distribution of future states of the process, given the present state, depends only upon the current state, i. ... In the mathematics of probability, a stochastic process is a random function. ... In probability theory, an event happens almost surely (a. ...

then classical Wiener measure γ is the law of the process B. 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. ...

Alternatively, one may use the abstract Wiener space construction, in which classical Wiener measure γ is the radonification of the canonical Gaussian cylinder set measure on the Cameron-Martin Hilbert space corresponding to C₀. 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. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ...

Classical Wiener measure is a Gaussian measure: in particular, it is a strictly positive probability measure. In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space , closely related to the normal distribution in statistics. ... In mathematics, strict positivity is a concept in measure theory. ...

Given classical Wiener measure γ on C₀, the product measure is a probability measure on C, where γⁿ denotes the standard Gaussian measure on . In mathematics, given two measurable spaces and measures on them, one can obtain the product measurable space and the product measure on that space. ... In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space , closely related to the normal distribution in statistics. ...

See also

• Skorokhod space, a generalization of classical Wiener space, which allows functions to be discontinuous.

In mathematics, a càdlàg function is a function defined on the real numbers (or a subset of them) that is everywhere right-continuous and has left limits everywhere. ...

References

• Billingsley, Patrick (1995). Probability and Measure. John Wiley & Sons, Inc., New York. ISBN 0-471-00710-2.