# Wiener space, abstract

The probability distribution of a Brownian motion $\{ {B ( t ) } : {t \geq 0 } \}$ is a Gaussian measure (cf. also Constructive quantum field theory) that can be supported by the space $C = C [ 0, \infty )$ of continuous functions. For this reason, $C$ is also called the classical Wiener space. This notion can be generalized to a Banach space on which a Gaussian measure can be introduced.

Advanced stochastic analysis can be carried out on a Wiener space. The analysis was initiated by P. Lévy and N. Wiener, and a systematic development was made by R.H. Cameron and W.T. Martin, I.E. Segal, L. Gross, K. Itô, and others.

Following [a2], one can introduce the notion of an abstract Wiener space, which is a generalization of the classical Wiener space.

Let $H$ be a real separable Hilbert space with norm $\| \cdot \|$. On $H$ one introduces the weak Gaussian distribution $\nu$ in such a way that on any finite-, say $n$-, dimensional subspace $K$ of $H$ the restriction of $\nu$ to $K$ is the $n$- dimensional standard Gaussian distribution. In fact, $\nu$ may be called the weak white noise measure. A semi-norm (or norm) $\| \cdot \| _ {1}$ on $H$ is called a measurable norm if for any positive $\epsilon$ there exists a finite-dimensional projection operator $P _ {0}$ such that for any finite-dimensional projection operator $P$ orthogonal to $P _ {0}$ the inequality $\nu \{ x : {\| {Px } \| _ {1} > \epsilon } \} < \epsilon$ holds.

Now, let $\| x \| _ {1}$ be a measurable norm on $H$ and let $B$ be the completion of $H$ with respect to this norm (cf. Complete space). Then $B$ is a Banach space. Let ${\mathcal R}$ be the $\sigma$- ring generated by the cylinder subsets of $B$( cf. Cylinder set). For a cylinder set measure $\mu$ on ${\mathcal R}$ induced by the Gaussian distribution on $H$, the measure $\mu$ is countably additive on ${\mathcal R}$. Therefore, taking the $\sigma$- field ${\mathcal B}$ generated by ${\mathcal R}$, a measure space $( H, {\mathcal B}, \mu )$ is obtained.

The classical definition of an abstract Wiener space is given as follows. First, take a Hilbert space $H$ with norm $\| \cdot \|$ and take a measurable norm $\| \cdot \| _ {1}$, to obtain a Banach space $B$. The injection mapping from $H$ into $B$ is denoted by $i$. Then the triple $( i,H,B )$ is called an abstract Wiener space. This means that a weak measure on $H$ can be extended to a completely additive measure supported by $B$. A stochastic analysis can be developed for this latter measure (see [a4]).

One of the developments of the notion of an abstract Wiener space is that of a rigged Hilbert space, due to I.M. Gel'fand and N.Ya. Vilenkin (see [a1]). Let $H$ be a real Hilbert space and let $\Phi$ be a countably Hilbert nuclear space that is continuously imbedded in $H$. The dual space $\Phi ^ {*}$ of $\Phi$ gives rise to the rigged Hilbert space

$$\Phi \subset H \subset \Phi ^ {*} .$$

Given a characteristic functional $C ( \xi )$, $\xi \in \Phi$, that is, $C ( \xi )$ is continuous in $\xi$, positive definite and $C ( 0 ) = 1$, there exists a countably additive probability measure $\mu$ in $\Phi ^ {*}$ such that

$$C ( \xi ) = \int\limits _ {\Phi ^ {*} } { { \mathop{\rm exp} } [ i \left \langle {x, \xi } \right \rangle ] } {d \mu ( x ) } .$$

A typical example of a rigged Hilbert space is the triple consisting of the Hilbert space $L _ {2} ( \mathbf R )$, the Schwartz space $S$ and the space $S ^ {*}$ of tempered distributions (cf. Generalized function). White noise is also an important example of $\mu$ on $S ^ {*}$; it has characteristic functional $C ( \xi ) = { \mathop{\rm exp} } [ - { {\| \xi \| ^ {2} } / 2 } ]$. The analysis on the function space with the white noise measure is well developed (see [a3]).

#### References

 [a1] I.M. Gel'fand, N.Ya. Vilenkin, "Generalized functions 4: applications of harmonic analysis" , Acad. Press (1964) (In Russian) [a2] L. Gross, "Abstract Wiener spaces" , Proc. 5th Berkeley Symp. Math. Stat. Probab. , 2 (1965) pp. 31–42 [a3] T. Hida, H.H. Kuo, J. Potthoff, L. Streit, "White noise. An infinite dimensional calculus" , Kluwer Acad. Publ. (1993) [a4] H.H. Kuo, "Gaussian measures in Banach spaces" , Lecture Notes in Mathematics , 463 , Springer (1975)
How to Cite This Entry:
Wiener space, abstract. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wiener_space,_abstract&oldid=49223
This article was adapted from an original article by T. Hida (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article