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The concept of an abstract Wiener space is a mathematical construction developed by Leonard Gross to understand the structure of
Gaussian measure In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space \mathbb^n, closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are na ...
s on infinite-dimensional spaces. The construction emphasizes the fundamental role played by the Cameron–Martin space. The
classical Wiener space In mathematics, classical Wiener space is the collection of all continuous functions on a given domain (usually a subinterval of the real line), taking values in a metric space (usually ''n''-dimensional Euclidean space). Classical Wiener space ...
is the prototypical example. The structure theorem for Gaussian measures states that all Gaussian measures can be represented by the abstract Wiener space construction.


Motivation

Let H be a real
Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
, assumed to be infinite dimensional and separable. In the physics literature, one frequently encounters integrals of the form :\frac\int_H f(v) e^ Dv, where Z is supposed to be a normalization constant and where Dv is supposed to be the non-existent Lebesgue measure on H. Such integrals arise, notably, in the context of the Euclidean path-integral formulation of quantum field theory. At a mathematical level, such an integral cannot be interpreted as integration against a measure on the original Hilbert space H. On the other hand, suppose B is a Banach space that contains H as a
dense Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be use ...
subspace. If B is "sufficiently larger" than H, then the above integral can be interpreted as integration against a well-defined (Gaussian) measure on B. In that case, the pair (H,B) is referred to as an abstract Wiener space. The prototypical example is the classical Wiener space, in which H is the Hilbert space of real-valued functions b on an interval ,T/math> having first derivative in L^2 and satisfying b(0) = 0, with the norm being given by :\left\Vert b\right\Vert^2 = \int_0^T b'(t)^2\,dt. In that case, B may be taken to be the Banach space of continuous functions on ,T/math> with the
supremum norm In mathematical analysis, the uniform norm (or ) assigns, to real- or complex-valued bounded functions defined on a set , the non-negative number :\, f\, _\infty = \, f\, _ = \sup\left\. This norm is also called the , the , the , or, when t ...
. In this case, the measure on B is the
Wiener measure In mathematics, the Wiener process (or Brownian motion, due to its historical connection with the physical process of the same name) is a real-valued continuous-time stochastic process discovered by Norbert Wiener. It is one of the best know ...
describing
Brownian motion Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
starting at the origin. The original subspace H\subset B is called the Cameron–Martin space, which forms a set of measure zero with respect to the Wiener measure. What the preceding example means is that we have a ''formal'' expression for the Wiener measure given by d\mu(b)=\frac \exp\left\\,Db. Although this formal expression ''suggests'' that the Wiener measure should live on the space of paths for which \int_0^T b'(t)^2\,dt < \infty, this is not actually the case, as sample Brownian paths are known to be
almost surely In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (with respect to the probability measure). In other words, the set of outcomes on which the event does not occur ha ...
nowhere differentiable, though it can be generalized to random measures as tempered distributions through the characteristic function as the white noise measure. Gross's abstract Wiener space construction abstracts the situation for the classical Wiener space and provides a necessary and sufficient (if sometimes difficult to check) condition for the Gaussian measure to exist on B. Although the Gaussian measure \mu lives on B rather than H, it is the geometry of H rather than B that controls the properties of \mu. As Gross himself puts it (adapted to our notation), "However, it only became apparent with the work of I.E. Segal dealing with the normal distribution on a real Hilbert space, that the role of the Hilbert space H was indeed central, and that in so far as analysis on B is concerned, the role of B itself was auxiliary for many of Cameron and Martin's theorems, and in some instances even unnecessary." One of the appealing features of Gross's abstract Wiener space construction is that it takes H as the starting point and treats B as an auxiliary object. Although the formal expressions for \mu appearing earlier in this section are purely formal, physics-style expressions, they are very useful in helping to understand properties of \mu. Notably, one can easily use these expressions to derive the (correct!) formula for the density of the translated measure d\mu(b+h) relative to d\mu(b), for h\in H. (See the
Cameron–Martin theorem In mathematics, the Cameron–Martin theorem or Cameron–Martin formula (named after Robert Horton Cameron and W. T. Martin) is a theorem of measure theory that describes how abstract Wiener measure changes under translation by certain eleme ...
.)


Mathematical description


Cylinder set measure on

Let H be a Hilbert space defined over the real numbers, assumed to be infinite dimensional and separable. A
cylinder set In mathematics, the cylinder sets form a basis of the product topology on a product of sets; they are also a generating family of the cylinder σ-algebra. General definition Given a collection S of sets, consider the Cartesian product X = \prod ...
in H is a set defined in terms of the values of a finite collection of linear functionals on H. Specifically, suppose \phi_1,\ldots,\phi_n are continuous linear functionals on H and E is a
Borel set In mathematics, a Borel set is any subset of a topological space that can be formed from its open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets ...
in \R^n. Then we can consider the set C = \left\. Any set of this type is called a cylinder set. The collection of all cylinder sets forms an algebra of sets in H, called the cylindrical algebra. Note that this algebra is ''not'' a \sigma-algebra. There is a natural way of defining a "measure" on cylinder sets, as follows. By the
Riesz representation theorem The Riesz representation theorem, sometimes called the Riesz–Fréchet representation theorem after Frigyes Riesz and Maurice René Fréchet, establishes an important connection between a Hilbert space and its continuous dual space. If the un ...
, the linear functionals \phi_1, \ldots, \phi_n are given as the inner product with vectors v_1, \ldots, v_n in H. In light of the Gram–Schmidt procedure, it is harmless to assume that v_1, \ldots, v_n are orthonormal. In that case, we can associate to the above-defined cylinder set C the measure of E with respect to the standard Gaussian measure on \mathbb R^n. That is, we define \mu(C)=(2\pi)^\int_e^\,dx, where dx is the standard Lebesgue measure on \R^n. Because of the product structure of the standard Gaussian measure on \R^n, it is not hard to show that \mu is well defined. That is, although the same set C can be represented as a cylinder set in more than one way, the value of \mu(C) is always the same.


Nonexistence of the measure on

The set functional \mu is called the standard Gaussian cylinder set measure on H. Assuming (as we do) that H is infinite dimensional, \mu ''does not'' extend to a countably additive measure on the \sigma-algebra generated by the collection of cylinder sets in H (that is, it does not extend to the cylindrical σ-algebra generated by the cylinder algebra.) One can understand the difficulty by considering the behavior of the standard Gaussian measure on \R^n, given by (2\pi)^ e^\,dx. The expectation value of the squared norm with respect to this measure is computed as an elementary
Gaussian integral The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function f(x) = e^ over the entire real line. Named after the German mathematician Carl Friedrich Gauss, the integral is \int_^\infty e^\,dx = \s ...
as (2\pi)^ \int_ \Vert x\Vert^2 e^ \,dx = (2\pi)^ \sum_^n \int_\R x_i^2 e^ \, dx_i = n. That is, the typical distance from the origin of a vector chosen randomly according to the standard Gaussian measure on \R^n is \sqrt n. As n tends to infinity, this typical distance tends to infinity, indicating that there is no well-defined "standard Gaussian" measure on H. (The typical distance from the origin would be infinite, so that the measure would not actually live on the space H.)


Existence of the measure on

Now suppose that B is a separable Banach space and that i:H\rightarrow B is an
injective In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, impl ...
continuous linear map whose image is dense in B. It is then harmless (and convenient) to identify H with its image inside B and thus regard H as a dense subset of B. We may then construct a cylinder set measure on B by defining the measure of a cylinder set C\subset B to be the previously defined cylinder set measure of C\cap H, which is a cylinder set in H. The idea of the abstract Wiener space construction is that if B is sufficiently bigger than H, then the cylinder set measure on B, unlike the cylinder set measure on H, will extend to a countably additive measure on the generated \sigma-algebra. The original paper of Gross gives a necessary and sufficient condition on B for this to be the case. The measure on B is called a Gaussian measure and the subspace H\subset B is called the Cameron–Martin space. It is important to emphasize that H forms a set of measure zero inside B, emphasizing that the Gaussian measure lives only on B and not on H. The upshot of this whole discussion is that Gaussian integrals of the sort described in the motivation section do have a rigorous mathematical interpretation, but they do not live on the space whose norm occurs in the exponent of the formal expression. Rather, they live on some larger space.


Universality of the construction

The abstract Wiener space construction is not simply one method of building Gaussian measures. Rather, ''every'' Gaussian measure on an infinite-dimensional Banach space occurs in this way. (See the structure theorem for Gaussian measures.) That is, given a Gaussian measure \mu on an infinite-dimensional, separable Banach space (over \mathbb R), one can identify a Cameron–Martin subspace H\subset B, at which point the pair (H,B) becomes an abstract Wiener space and \mu is the associated Gaussian measure.


Properties

* \mu is a
Borel measure In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. ...
: it is defined on the Borel σ-algebra generated by the
open subsets In mathematics, an open set is a generalization of an open interval in the real line. In a metric space (a set with a distance defined between every two points), an open set is a set that, with every point in it, contains all points of the metr ...
of ''B''. * \mu is a
Gaussian measure In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space \mathbb^n, closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are na ...
in the sense that ''f''(\mu) is a Gaussian measure on R for every
linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear mapIn some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars from a vector space to its field of ...
, . * Hence, \mu is strictly positive and locally finite. * The behaviour of \mu under
translation Translation is the communication of the semantics, meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The English la ...
is described by the
Cameron–Martin theorem In mathematics, the Cameron–Martin theorem or Cameron–Martin formula (named after Robert Horton Cameron and W. T. Martin) is a theorem of measure theory that describes how abstract Wiener measure changes under translation by certain eleme ...
. * Given two abstract Wiener spaces and , one can show that \gamma_=\gamma_1\otimes\gamma_2. In full: (i_1 \times i_2)_* (\mu^) = (i_1)_* \left( \mu^ \right) \otimes (i_2)_* \left( \mu^ \right), i.e., the abstract Wiener measure \mu_ on the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...
is the product of the abstract Wiener measures on the two factors and . * If ''H'' (and ''B'') are infinite dimensional, then the image of ''H'' has
measure zero In mathematical analysis, a null set is a Lebesgue measurable set of real numbers that has Lebesgue measure, measure zero. This can be characterized as a set that can be Cover (topology), covered by a countable union of Interval (mathematics), ...
. This fact is a consequence of
Kolmogorov's zero–one law In probability theory, Kolmogorov's zero–one law, named in honor of Andrey Nikolaevich Kolmogorov, specifies that a certain type of event, namely a ''tail event of independent σ-algebras'', will either almost surely happen or almost su ...
. * The inclusion map i\colon H\hookrightarrow E is an example of a \gamma- radonyfing map, since the pushforward measure i_*\mu is a
Radon measure In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the -algebra of Borel sets of a Hausdorff topological space that is finite on all compact sets, outer regular on all Borel sets, and ...
. Gross originally formulated a necessary and sufficient condition for \gamma-radonification in terms of the measurable seminorms.


Example: Classical Wiener space

The prototypical example of an abstract Wiener space takes the space B to be
classical Wiener space In mathematics, classical Wiener space is the collection of all continuous functions on a given domain (usually a subinterval of the real line), taking values in a metric space (usually ''n''-dimensional Euclidean space). Classical Wiener space ...
, the space of continuous paths. The subspace H is given by : \begin H &:= L_^ (
, T The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
\mathbb^) \\ &:= \ \end with
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, ofte ...
given by :\langle \sigma_1, \sigma_2 \rangle_ := \int_0^T \langle \dot_1 (t), \dot_2 (t) \rangle_ \, dt. The classical Wiener space B is then the space of continuous maps of ,T/math> into \mathbb R^n starting at 0, with the
uniform norm In mathematical analysis, the uniform norm (or ) assigns, to real- or complex-valued bounded functions defined on a set , the non-negative number :\, f\, _\infty = \, f\, _ = \sup\left\. This norm is also called the , the , the , or, when t ...
. In this case, the Gaussian measure \mu is the
Wiener measure In mathematics, the Wiener process (or Brownian motion, due to its historical connection with the physical process of the same name) is a real-valued continuous-time stochastic process discovered by Norbert Wiener. It is one of the best know ...
, which describes
Brownian motion Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
in \mathbb R^n, starting from the origin. The general result that H forms a set of measure zero with respect to \mu in this case reflects the roughness of the typical Brownian path, which is known to be
nowhere differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
. This contrasts with the assumed differentiability of the paths in H.


See also

* * * * * * *


References

* (See section 1.1) * * * {{Analysis in topological vector spaces Measure theory Stochastic processes