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)