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The [[Probability distribution|probability distribution]] of a [[Brownian motion|Brownian motion]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w1100701.png" /> is a Gaussian measure (cf. also [[Constructive quantum field theory|Constructive quantum field theory]]) that can be supported by the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w1100702.png" /> of continuous functions. For this reason, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w1100703.png" /> is also called the classical Wiener space. This notion can be generalized to a [[Banach space|Banach space]] on which a Gaussian measure can be introduced.
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The [[Probability distribution|probability distribution]] of a [[Brownian motion|Brownian motion]] $  \{ {B ( t ) } : {t \geq  0 } \} $
 +
is a Gaussian measure (cf. also [[Constructive quantum field theory|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|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.
 
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.
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Following [[#References|[a2]]], one can introduce the notion of an abstract Wiener space, which is a generalization of the classical Wiener space.
 
Following [[#References|[a2]]], one can introduce the notion of an abstract Wiener space, which is a generalization of the classical Wiener space.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w1100704.png" /> be a real separable [[Hilbert space|Hilbert space]] with norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w1100705.png" />. On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w1100706.png" /> one introduces the weak Gaussian distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w1100707.png" /> in such a way that on any finite-, say <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w1100708.png" />-, dimensional subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w1100709.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007010.png" /> the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007011.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007012.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007013.png" />-dimensional standard Gaussian distribution. In fact, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007014.png" /> may be called the weak white noise measure. A semi-norm (or norm) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007015.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007016.png" /> is called a measurable norm if for any positive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007017.png" /> there exists a finite-dimensional projection operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007018.png" /> such that for any finite-dimensional projection operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007019.png" /> orthogonal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007020.png" /> the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007021.png" /> holds.
+
Let $  H $
 +
be a real separable [[Hilbert space|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007022.png" /> be a measurable norm on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007023.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007024.png" /> be the completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007025.png" /> with respect to this norm (cf. [[Complete space|Complete space]]). Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007026.png" /> is a Banach space. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007027.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007028.png" />-ring generated by the cylinder subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007029.png" /> (cf. [[Cylinder set|Cylinder set]]). For a cylinder set measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007030.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007031.png" /> induced by the Gaussian distribution on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007032.png" />, the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007033.png" /> is countably additive on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007034.png" />. Therefore, taking the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007035.png" />-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007036.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007037.png" />, a [[Measure space|measure space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007038.png" /> is obtained.
+
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|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|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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007039.png" /> with norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007040.png" /> and take a measurable norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007041.png" />, to obtain a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007042.png" />. The injection mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007043.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007044.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007045.png" />. Then the triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007046.png" /> is called an abstract Wiener space. This means that a weak measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007047.png" /> can be extended to a completely additive measure supported by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007048.png" />. A stochastic analysis can be developed for this latter measure (see [[#References|[a4]]]).
+
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 [[#References|[a4]]]).
  
One of the developments of the notion of an abstract Wiener space is that of a [[Rigged Hilbert space|rigged Hilbert space]], due to I.M. Gel'fand and N.Ya. Vilenkin (see [[#References|[a1]]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007049.png" /> be a real Hilbert space and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007050.png" /> be a countably Hilbert [[Nuclear space|nuclear space]] that is continuously imbedded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007051.png" />. The dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007052.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007053.png" /> gives rise to the rigged Hilbert space
+
One of the developments of the notion of an abstract Wiener space is that of a [[Rigged Hilbert space|rigged Hilbert space]], due to I.M. Gel'fand and N.Ya. Vilenkin (see [[#References|[a1]]]). Let $  H $
 +
be a real Hilbert space and let $  \Phi $
 +
be a countably Hilbert [[Nuclear space|nuclear space]] that is continuously imbedded in $  H $.  
 +
The dual space $  \Phi  ^ {*} $
 +
of $  \Phi $
 +
gives rise to the rigged Hilbert space
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007054.png" /></td> </tr></table>
+
$$
 +
\Phi \subset  H \subset  \Phi  ^ {*} .
 +
$$
  
Given a characteristic functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007056.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007057.png" /> is continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007058.png" />, positive definite and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007059.png" />, there exists a countably additive probability measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007060.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007061.png" /> such that
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007062.png" /></td> </tr></table>
+
$$
 +
C ( \xi ) = \int\limits _ {\Phi  ^ {*} } { { \mathop{\rm exp} } [ i \left \langle  {x, \xi } \right \rangle ] }  {d \mu ( x ) } .
 +
$$
  
A typical example of a rigged Hilbert space is the triple consisting of the Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007063.png" />, the Schwartz space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007064.png" /> and the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007065.png" /> of tempered distributions (cf. [[Generalized function|Generalized function]]). [[White noise|White noise]] is also an important example of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007066.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007067.png" />; it has characteristic functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110070/w11007068.png" />. The analysis on the function space with the white noise measure is well developed (see [[#References|[a3]]]).
+
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|Generalized function]]). [[White noise|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 [[#References|[a3]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.M. Gel'fand,  N.Ya. Vilenkin,  "Generalized functions 4: applications of harmonic analysis" , Acad. Press  (1964)  (In Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Gross,  "Abstract Wiener spaces" , ''Proc. 5th Berkeley Symp. Math. Stat. Probab.'' , '''2'''  (1965)  pp. 31–42</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  T. Hida,  H.H. Kuo,  J. Potthoff,  L. Streit,  "White noise. An infinite dimensional calculus" , Kluwer Acad. Publ.  (1993)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H.H. Kuo,  "Gaussian measures in Banach spaces" , ''Lecture Notes in Mathematics'' , '''463''' , Springer  (1975)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.M. Gel'fand,  N.Ya. Vilenkin,  "Generalized functions 4: applications of harmonic analysis" , Acad. Press  (1964)  (In Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Gross,  "Abstract Wiener spaces" , ''Proc. 5th Berkeley Symp. Math. Stat. Probab.'' , '''2'''  (1965)  pp. 31–42</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  T. Hida,  H.H. Kuo,  J. Potthoff,  L. Streit,  "White noise. An infinite dimensional calculus" , Kluwer Acad. Publ.  (1993)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H.H. Kuo,  "Gaussian measures in Banach spaces" , ''Lecture Notes in Mathematics'' , '''463''' , Springer  (1975)</TD></TR></table>

Latest revision as of 08:29, 6 June 2020


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=15560
This article was adapted from an original article by T. Hida (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article