The concept of anabstract Wiener space is a mathematical construction developed byLeonard Gross to understand the structure ofGaussian measures on infinite-dimensional spaces. The construction emphasizes the fundamental role played by theCameron–Martin space. Theclassical Wiener space is the prototypical example.

Thestructure theorem for Gaussian measures states thatall Gaussian measures can be represented by the abstract Wiener space construction.

Let${\displaystyle H}$ be a realHilbert space, assumed to be infinite dimensional andseparable. In the physics literature, one frequently encounters integrals of the form

$\frac{1}{Z}{\int }_{H}f(v)e^{-\frac{1}{2}\Vert v{\Vert }^2}Dv,$

where${\displaystyle Z}$ is supposed to be a normalization constant and where${\displaystyle Dv}$ is supposed to be thenon-existent Lebesgue measure on${\displaystyle H}$. Such integrals arise, notably, in the context of theEuclidean path-integral formulation of quantum field theory. At a mathematical level, such an integral cannot be interpreted as integration against ameasure on the original Hilbert space${\displaystyle H}$. On the other hand, suppose${\displaystyle B}$ is aBanach space that contains${\displaystyle H}$ as adense subspace. If${\displaystyle B}$ is "sufficiently larger" than${\displaystyle H}$, then the above integral can be interpreted as integration against a well-defined (Gaussian) measure on${\displaystyle B}$. In that case, the pair${\displaystyle (H,B)}$ is referred to as an abstract Wiener space.

The prototypical example is the classical Wiener space, in which${\displaystyle H}$ is the Hilbert space of real-valued functions${\displaystyle b}$ on an interval${\displaystyle [0,T]}$ having first derivative in${\displaystyle L^2}$ and satisfying${\displaystyle b(0)=0}$, with the norm being given by

${\displaystyle {\Vert b\Vert }^2={\int }_0^T{b}^{\prime }(t{)}^{2}dt.}$

In that case,${\displaystyle B}$ may be taken to be the Banach space of continuous functions on${\displaystyle [0,T]}$ with thesupremum norm. In this case, the measure on${\displaystyle B}$ is theWiener measure describingBrownian motion starting at the origin. The original subspace${\displaystyle H\subset B}$ is called theCameron–Martin space, which forms a set of measure zero with respect to the Wiener measure.

What the preceding example means is that we have aformal expression for the Wiener measure given by$${\displaystyle d\mu (b)=\frac{1}{Z}\exp \left\{-\frac{1}{2}{\int }_0^T{b}^{\prime }(t{)}^{2}dt\right\}Db.}$$

Although this formal expressionsuggests that the Wiener measure should live on the space of paths for which, this is not actually the case, as sample Brownian paths are known to bealmost surely nowhere differentiable, though it can be generalized to random measures astempered distributions through the characteristic function as thewhite 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${\displaystyle B}$. Although the Gaussian measure${\displaystyle \mu }$ lives on${\displaystyle B}$ rather than${\displaystyle H}$, it is the geometry of${\displaystyle H}$ rather than${\displaystyle B}$ that controls the properties of${\displaystyle \mu }$. As Gross himself puts it^[1] (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${\displaystyle H}$ was indeed central, and that in so far as analysis on${\displaystyle B}$ is concerned, the role of${\displaystyle B}$ 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${\displaystyle H}$ as the starting point and treats${\displaystyle B}$ as an auxiliary object.

Although the formal expressions for${\displaystyle \mu }$ appearing earlier in this section are purely formal, physics-style expressions, they are very useful in helping to understand properties of${\displaystyle \mu }$. Notably, one can easily use these expressions to derive the (correct!) formula for the density of the translated measure${\displaystyle d\mu (b+h)}$ relative to${\displaystyle d\mu (b)}$, for${\displaystyle h\in H}$. (See theCameron–Martin theorem.)

Let${\displaystyle H}$ be a Hilbert space defined over the real numbers, assumed to be infinite dimensional and separable. Acylinder set in${\displaystyle H}$ is a set defined in terms of the values of a finite collection of linear functionals on${\displaystyle H}$. Specifically, suppose${\displaystyle {\varphi }_1,\dots ,{\varphi }_n}$ are continuous linear functionals on${\displaystyle H}$ and${\displaystyle E}$ is aBorel set in${\displaystyle {\mathbb{R}}^n}$. Then we can consider the set$${\displaystyle C=\left\{v\in H\mid ({\varphi }_1(v),\dots ,{\varphi }_n(v))\in E\right\}.}$$

Any set of this type is called a cylinder set. The collection of all cylinder sets forms an algebra of sets in${\displaystyle H,}$ called thecylindrical algebra. Note that this algebra isnot a${\displaystyle \sigma }$-algebra.

There is a natural way of defining a "measure" on cylinder sets, as follows. By theRiesz representation theorem, the linear functionals${\displaystyle {\varphi }_1,\dots ,{\varphi }_n}$ are given as the inner product with vectors${\displaystyle v_1,\dots ,v_n}$ in${\displaystyle H}$. In light of theGram–Schmidt procedure, it is harmless to assume that${\displaystyle v_1,\dots ,v_n}$ are orthonormal. In that case, we can associate to the above-defined cylinder set${\displaystyle C}$ the measure of${\displaystyle E}$ with respect to the standard Gaussian measure on${\displaystyle {\mathbb{R}}^n}$. That is, we define$${\displaystyle \mu (C)=(2\pi {)}^{-n/2}{\int }_{E\subset {\mathbb{R}}^n}{e}^{-\Vert x{\Vert }^2/2}dx,}$$where${\displaystyle dx}$ is the standard Lebesgue measure on${\displaystyle {\mathbb{R}}^n}$. Because of the product structure of the standard Gaussian measure on${\displaystyle {\mathbb{R}}^n}$, it is not hard to show that${\displaystyle \mu }$ is well defined. That is, although the same set${\displaystyle C}$ can be represented as a cylinder set in more than one way, the value of${\displaystyle \mu (C)}$ is always the same.

The set functional${\displaystyle \mu }$ is called the standard Gaussiancylinder set measure on${\displaystyle H}$. Assuming (as we do) that${\displaystyle H}$ is infinite dimensional,${\displaystyle \mu }$does not extend to a countably additive measure on the${\displaystyle \sigma }$-algebra generated by the collection of cylinder sets in${\displaystyle H}$ (that is, it does not extend to thecylindrical σ-algebra generated by the cylinder algebra.) One can understand the difficulty by considering the behavior of the standard Gaussian measure on${\displaystyle {\mathbb{R}}^n,}$ given by$${\displaystyle (2\pi {)}^{-n/2}{e}^{-\Vert x{\Vert }^2/2}dx.}$$

The expectation value of the squared norm with respect to this measure is computed as an elementaryGaussian integral as$${\displaystyle (2\pi {)}^{-n/2}{\int }_{{\mathbb{R}}^n}\Vert x{\Vert }^2{e}^{-\Vert x{\Vert }^2/2}dx=(2\pi {)}^{-1/2}\sum _{i=1}^n{\int }_{\mathbb{R}}{x}_i^2{e}^{-x_i^2/2}d{x}_i=n.}$$

That is, the typical distance from the origin of a vector chosen randomly according to the standard Gaussian measure on${\displaystyle {\mathbb{R}}^n}$ is${\displaystyle \sqrt{n}.}$ As${\displaystyle n}$ tends to infinity, this typical distance tends to infinity, indicating that there is no well-defined "standard Gaussian" measure on${\displaystyle H}$. (The typical distance from the origin would be infinite, so that the measure would not actually live on the space${\displaystyle H}$.)

Now suppose that${\displaystyle B}$ is a separable Banach space and that${\displaystyle i:H\to B}$ is aninjectivecontinuous linear map whose image is dense in${\displaystyle B}$. It is then harmless (and convenient) to identify${\displaystyle H}$ with its image inside${\displaystyle B}$ and thus regard${\displaystyle H}$ as a dense subset of${\displaystyle B}$. We may then construct a cylinder set measure on${\displaystyle B}$ by defining the measure of a cylinder set${\displaystyle C\subset B}$ to be the previously defined cylinder set measure of${\displaystyle C\cap H}$, which is a cylinder set in${\displaystyle H}$.

The idea of the abstract Wiener space construction is that if${\displaystyle B}$ is sufficiently bigger than${\displaystyle H}$, then the cylinder set measure on${\displaystyle B}$, unlike the cylinder set measure on${\displaystyle H}$, will extend to a countably additive measure on the generated${\displaystyle \sigma }$-algebra. The original paper of Gross^[2] gives a necessary and sufficient condition on${\displaystyle B}$ for this to be the case. The measure on${\displaystyle B}$ is called aGaussian measure and the subspace${\displaystyle H\subset B}$ is called theCameron–Martin space. It is important to emphasize that${\displaystyle H}$ forms a set of measure zero inside${\displaystyle B}$, emphasizing that the Gaussian measure lives only on${\displaystyle B}$ and not on${\displaystyle H}$.

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.

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 thestructure theorem for Gaussian measures.) That is, given a Gaussian measure${\displaystyle \mu }$ on an infinite-dimensional, separable Banach space (over${\displaystyle \mathbb{R}}$), one can identify aCameron–Martin subspace${\displaystyle H\subset B}$, at which point the pair${\displaystyle (H,B)}$ becomes an abstract Wiener space and${\displaystyle \mu }$ is the associated Gaussian measure.

• ${\displaystyle \mu }$ is aBorel measure: it is defined on theBorel σ-algebra generated by theopen subsets ofB.
• ${\displaystyle \mu }$ is aGaussian measure in the sense thatf_∗(${\displaystyle \mu }$) is a Gaussian measure onR for everylinear functionalf ∈B^∗,f ≠ 0.
• Hence,${\displaystyle \mu }$ is strictly positive and locally finite.
• The behaviour of${\displaystyle \mu }$ undertranslation is described by theCameron–Martin theorem.
• Given two abstract Wiener spacesi₁ :H₁ →B₁ andi₂ :H₂ →B₂, one can show that${\displaystyle {\gamma }_{12}={\gamma }_1\otimes {\gamma }_2}$. In full:$${\displaystyle (i_1\times i_2{)}_{\ast }({\mu }^{H_1\times H_2})=(i_1{)}_{\ast }\left({\mu }^{H_1}\right)\otimes (i_2{)}_{\ast }\left({\mu }^{H_2}\right),}$$ i.e., the abstract Wiener measure${\displaystyle {\mu }_{12}}$ on theCartesian productB₁ ×B₂ is the product of the abstract Wiener measures on the two factorsB₁ andB₂.
• IfH (andB) are infinite dimensional, then the image ofH hasmeasure zero. This fact is a consequence ofKolmogorov's zero–one law.
• The inclusion map${\displaystyle i:H\hookrightarrow E}$ is an example of a${\displaystyle \gamma }$-radonifying map, since the pushforward measure${\displaystyle i_{\ast }\mu }$ is aRadon measure. Gross originally formulated a necessary and sufficient condition for${\displaystyle \gamma }$-radonification in terms of the measurable seminorms.

The prototypical example of an abstract Wiener space takes the space${\displaystyle B}$ to beclassical Wiener space, the space of continuouspaths. The subspace${\displaystyle H}$ is given by

withinner product given by

${\displaystyle ⟨{\sigma }_1,{\sigma }_2{⟩}_{L_0^{2,1}}:={\int }_0^T⟨{\dot{\sigma }}_1(t),{\dot{\sigma }}_2(t){⟩}_{{\mathbb{R}}^n}dt.}$

The classical Wiener space${\displaystyle B}$ is then the space of continuous maps of${\displaystyle [0,T]}$ into${\displaystyle {\mathbb{R}}^n}$ starting at 0, with theuniform norm. In this case, the Gaussian measure${\displaystyle \mu }$ is theWiener measure, which describesBrownian motion in${\displaystyle {\mathbb{R}}^n}$, starting from the origin.

The general result that${\displaystyle H}$ forms a set of measure zero with respect to${\displaystyle \mu }$ in this case reflects the roughness of the typical Brownian path, which is known to benowhere differentiable. This contrasts with the assumed differentiability of the paths in${\displaystyle H}$.

• Besov measure – Generalization of the Gaussian measure using the Besov norm
• Cameron–Martin theorem – Theorem describing translation of Gaussian measures on Hilbert spaces
• Feldman–Hájek theorem – Theory in probability theory
• Gaussian probability space
• Malliavin calculus – Mathematical techniques used in probability theory and related fields
• Structure theorem for Gaussian measures – Mathematical theorem
• There is no infinite-dimensional Lebesgue measure – Mathematical folklorePages displaying short descriptions of redirect targets

• Bell, Denis R. (2006).The Malliavin calculus. Mineola, NY: Dover Publications Inc. p. x+113.ISBN 0-486-44994-7.MR 2250060. (See section 1.1)
• Gross, Leonard (1967). "Abstract Wiener spaces".Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), Vol. II: Contributions to Probability Theory, Part 1. Berkeley, Calif.: Univ. California Press. pp. 31–42.MR 0212152.
• Kuo, Hui-Hsiung (1975).Gaussian measures in Banach spaces. Berlin–New York: Springer. p. 232.ISBN 978-1419645808.
• Elworthy, David (2008),MA482 Stochastic Analysis(PDF), Lecture Notes, University of Warwick

• Measure theory
• Stochastic processes

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