Wiener space, abstract
The probability distribution of a Brownian motion is a Gaussian measure (cf. also Constructive quantum field theory) that can be supported by the space
of continuous functions. For this reason,
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 be a real separable Hilbert space with norm
. On
one introduces the weak Gaussian distribution
in such a way that on any finite-, say
-, dimensional subspace
of
the restriction of
to
is the
-dimensional standard Gaussian distribution. In fact,
may be called the weak white noise measure. A semi-norm (or norm)
on
is called a measurable norm if for any positive
there exists a finite-dimensional projection operator
such that for any finite-dimensional projection operator
orthogonal to
the inequality
holds.
Now, let be a measurable norm on
and let
be the completion of
with respect to this norm (cf. Complete space). Then
is a Banach space. Let
be the
-ring generated by the cylinder subsets of
(cf. Cylinder set). For a cylinder set measure
on
induced by the Gaussian distribution on
, the measure
is countably additive on
. Therefore, taking the
-field
generated by
, a measure space
is obtained.
The classical definition of an abstract Wiener space is given as follows. First, take a Hilbert space with norm
and take a measurable norm
, to obtain a Banach space
. The injection mapping from
into
is denoted by
. Then the triple
is called an abstract Wiener space. This means that a weak measure on
can be extended to a completely additive measure supported by
. 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 be a real Hilbert space and let
be a countably Hilbert nuclear space that is continuously imbedded in
. The dual space
of
gives rise to the rigged Hilbert space
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Given a characteristic functional ,
, that is,
is continuous in
, positive definite and
, there exists a countably additive probability measure
in
such that
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A typical example of a rigged Hilbert space is the triple consisting of the Hilbert space , the Schwartz space
and the space
of tempered distributions (cf. Generalized function). White noise is also an important example of
on
; it has characteristic functional
. 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) |
Wiener space, abstract. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wiener_space,_abstract&oldid=15560