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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w0978901.png" /> be a dense subspace of a separable Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w0978902.png" />. The triplet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w0978903.png" /> given by the injection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w0978904.png" /> is obtained by identifying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w0978905.png" /> with its dual, taking the dual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w0978906.png" />, and endowing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w0978907.png" />, the algebraic dual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w0978908.png" />, with the weak topology. For any real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w0978909.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789010.png" /> be the Hilbert space obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789011.png" /> by multiplying the norm on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789012.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789013.png" />.
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The dual of the symmetric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789014.png" />-fold tensor product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789015.png" /> is the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789016.png" /> of all homogeneous polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789017.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789018.png" />. The value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789019.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789020.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789021.png" />. Thus, for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789022.png" /> there is a triplet
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<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/w097/w097890/w09789023.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
Let  $  U $
 +
be a dense subspace of a separable Hilbert space  $  H $.
 +
The triplet  $  U \subset  H \subset  U  ^ {*} $
 +
given by the injection  $  i : U \rightarrow H $
 +
is obtained by identifying  $  H $
 +
with its dual, taking the dual of  $  i $,
 +
and endowing  $  U  ^ {*} $,
 +
the algebraic dual of  $  U $,
 +
with the weak topology. For any real  $  \lambda $,
 +
let  $  \lambda H $
 +
be the Hilbert space obtained from  $  H $
 +
by multiplying the norm on  $  H $
 +
by  $  \lambda $.
  
Taking the direct sum of the internal space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789024.png" /> and the Hilbert sum of the central spaces there results a triplet
+
The dual of the symmetric  $  k $-
 +
fold tensor product  $  S _ {k} ( U) $
 +
is the space $  \mathop{\rm Pol} _ {k} ( U) $
 +
of all homogeneous polynomials of degree  $  k $
 +
on  $  U $.
 +
The value of  $  F _ {k} \in  \mathop{\rm Pol} _ {k} ( U) $
 +
at  $  u \in U $
 +
is  $  F _ {k} ( u ) = \langle  F _ {k} , u ^ {\otimes k } \rangle _ {k! }  $.  
 +
Thus, for each  $  k $
 +
there is a triplet
  
<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/w097/w097890/w09789025.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
$$ \tag{a1 }
 +
S _ {k} ( U)  \subset  \sqrt k! S _ {k} ( H)  \subset    \mathop{\rm Pol} _ {k} ( U) .
 +
$$
 +
 
 +
Taking the direct sum of the internal space  $  S _ {k} ( U) $
 +
and the Hilbert sum of the central spaces there results a triplet
 +
 
 +
$$ \tag{a2 }
 +
S( U)  \subset    \mathop{\rm Fock} ( H)  \subset    \mathop\widehat{ {\rm Pol}}  ( U),
 +
$$
  
 
called dressed Fock space. The middle term is the usual [[Fock space|Fock space]]
 
called dressed Fock space. The middle term is the usual [[Fock space|Fock 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/w097/w097890/w09789026.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
+
$$ \tag{a3 }
 +
\mathop{\rm Fock} ( H)  = \oplus \sqrt k! S _ {k} ( H) .
 +
$$
  
The external space is the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789027.png" /> of all formal power series on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789028.png" />. The value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789029.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789030.png" /> of such an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789031.png" /> is defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789032.png" />, if this limit exists. For example, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789033.png" /> one has
+
The external space is the space $  \prod _ {k}  \mathop{\rm Pol} _ {k} ( U) $
 +
of all formal power series on $  U $.  
 +
The value $  F( u ) $
 +
at $  u \in U $
 +
of such an $  F \in  \mathop\widehat{ {\rm Pol}}  ( U) $
 +
is defined as $  \lim\limits _  \rightarrow  \sum _ {k=} 1  ^ {N} F _ {k} ( u ) $,  
 +
if this limit exists. For example, for any $  F = \sum F _ {k} \in  \mathop{\rm Fock} ( H) $
 +
one has
  
<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/w097/w097890/w09789034.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
+
$$ \tag{a4 }
 +
F( u ) =  \langle  F, e  ^ {u} \rangle ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789035.png" />.
+
where $  e  ^ {u} = \sum ( k!)  ^ {-} 1 u ^ {\otimes k } $.
  
 
A probabilized vector space is a structure
 
A probabilized vector space is a structure
  
<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/w097/w097890/w09789036.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5)</td></tr></table>
+
$$ \tag{a5 }
 +
( U \dots X \supset \Omega , {\mathsf P} )
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789038.png" /> are two spaces in duality and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789039.png" /> is linearly generated by the subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789040.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789041.png" />. This subset is endowed with a Polish (or Suslin) topology such that any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789042.png" /> defines a Borel function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789043.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789044.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789045.png" /> contains a countable subset separating the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789046.png" /> (so that the Borel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789047.png" />-field is generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789048.png" />). Finally, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789049.png" /> is a probability measure on this <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789050.png" />-field.
+
where $  U $
 +
and $  X $
 +
are two spaces in duality and $  X = \mathop{\rm span} ( \Omega ) $
 +
is linearly generated by the subset $  \Omega $
 +
of $  X $.  
 +
This subset is endowed with a Polish (or Suslin) topology such that any $  u \in U $
 +
defines a Borel function $  u( \omega ) = \langle  u , \omega \rangle $
 +
on $  \Omega $.  
 +
The space $  U $
 +
contains a countable subset separating the points of $  \Omega $(
 +
so that the Borel $  \sigma $-
 +
field is generated by $  U $).  
 +
Finally, $  {\mathsf P} $
 +
is a probability measure on this $  \sigma $-
 +
field.
  
Assume, moreover, that the space of cylindrical polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789051.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789052.png" />. Assume that the following bilinear form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789053.png" /> is a scalar product:
+
Assume, moreover, that the space of cylindrical polynomials $  P( \Omega ) = \mathop{\rm span} ( u ( \omega )  ^ {k} :  u \in U,  k = 0, 1, 2 , . . . ) $
 +
is dense in $  L _ {2} ( \Omega ) $.  
 +
Assume that the following bilinear form on $  U $
 +
is a scalar product:
  
<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/w097/w097890/w09789054.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a6)</td></tr></table>
+
$$ \tag{a6 }
 +
b( u , v)  = {\mathsf E} ( [ u ( \omega )- {\mathsf E} ( u ( \omega ))]
 +
[ v( \omega ) - {\mathsf E} ( v( \omega ))]) ,
 +
$$
  
and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789055.png" /> be the completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789056.png" />. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789057.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789058.png" /> denote the orthogonal projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789059.png" /> with range <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789060.png" />, the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789061.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789062.png" /> be the orthogonal complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789063.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789064.png" />. This space is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789065.png" />-th homogeneous chaos. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789066.png" /> is the Hilbert direct sum of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789067.png" />. One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789068.png" /> admits a decomposition in chaos if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789069.png" /> the following mapping is isometric:
+
and let $  H $
 +
be the completion of $  U $.  
 +
For any $  k > 0 $,  
 +
let $  \pi _ {k} $
 +
denote the orthogonal projection of $  L _ {2} ( \Omega ) $
 +
with range $  \overline{ {P _ {<} k ( \Omega ) }}\; $,  
 +
the closure of $  \mathop{\rm span} ( u ( \omega )  ^ {j} : u \in U,  j< k ) $.  
 +
Let $  KO _ {k} $
 +
be the orthogonal complement of $  \overline{ {P _ {<} k ( \Omega ) }}\; $
 +
in $  \overline{ {P _  \leq  k ( \Omega ) }}\; $.  
 +
This space is called the $  k $-
 +
th homogeneous chaos. The space $  L _ {2} ( \Omega ) $
 +
is the Hilbert direct sum of the $  KO _ {k} $.  
 +
One says that $  L _ {2} ( \Omega ) $
 +
admits a decomposition in chaos if for any $  k $
 +
the following mapping is isometric:
  
<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/w097/w097890/w09789070.png" /></td> </tr></table>
+
$$
 +
\sqrt k! S _ {k} ( H)  \supset  S _ {k} ( U)  \ni \
 +
Q  \mapsto ^ { {I _ k} }  Q - \pi _ {k} ( Q)  \in \
 +
KO _ {k}  \subset  L _ {2} ( \Omega ) .
 +
$$
  
The collection of these isometries for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789071.png" /> is an isometry <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789072.png" /> whose inverse
+
The collection of these isometries for $  k = 0, 1 \dots $
 +
is an isometry $  I $
 +
whose inverse
  
<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/w097/w097890/w09789073.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a7)</td></tr></table>
+
$$ \tag{a7 }
 +
L _ {2} ( \Omega )  \rightarrow ^ { {I  ^ {-}} 1 }  \mathop{\rm Fock} ( H) ,\ \
 +
f  \rightarrow  \widehat{f}  ,
 +
$$
  
extended to distributions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789074.png" />, is the starting point of distribution calculus on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789075.png" />. Because of (a4), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789076.png" /> is explicitly given by
+
extended to distributions on $  \Omega $,  
 +
is the starting point of distribution calculus on $  \Omega $.  
 +
Because of (a4), $  \widehat{f}  $
 +
is explicitly given by
  
<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/w097/w097890/w09789077.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a8)</td></tr></table>
+
$$ \tag{a8 }
 +
\widehat{f}  ( u )  = \langle  \widehat{f}  , e  ^ {u} \rangle  = {\mathsf E} [ f \epsilon  ^ {u} ] ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789078.png" />.
+
where $  \epsilon  ^ {u} = I  ^ {-} 1 ( e  ^ {u} ) $.
  
Decomposition in chaos was discovered by N. Wiener (in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789079.png" /> is Wiener space), [[#References|[a1]]]. Further contributions are due to Th.A. Dwyer and I. Segal ([[#References|[a2]]], [[#References|[a3]]]) and these have been important for constructive quantum field theory. K. Itô obtained a decomposition into chaos for Poisson probability spaces and interpreted <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097890/w09789080.png" /> as iterated stochastic integrals. For formula (a8), extended to distributions for Gaussian probability spaces, cf. [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]], [[#References|[a9]]], [[#References|[a10]]]. There are links with Malliavin calculus, [[#References|[a8]]].
+
Decomposition in chaos was discovered by N. Wiener (in the case $  \Omega $
 +
is Wiener space), [[#References|[a1]]]. Further contributions are due to Th.A. Dwyer and I. Segal ([[#References|[a2]]], [[#References|[a3]]]) and these have been important for constructive quantum field theory. K. Itô obtained a decomposition into chaos for Poisson probability spaces and interpreted $  I _ {k} ( f  ) $
 +
as iterated stochastic integrals. For formula (a8), extended to distributions for Gaussian probability spaces, cf. [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]], [[#References|[a9]]], [[#References|[a10]]]. There are links with Malliavin calculus, [[#References|[a8]]].
  
 
For more material cf. e.g. also [[#References|[a11]]], [[#References|[a12]]]; [[Wick product|Wick product]] and [[White noise analysis|White noise analysis]], and the references therein.
 
For more material cf. e.g. also [[#References|[a11]]], [[#References|[a12]]]; [[Wick product|Wick product]] and [[White noise analysis|White noise analysis]], and the references therein.

Revision as of 08:29, 6 June 2020


Let $ U $ be a dense subspace of a separable Hilbert space $ H $. The triplet $ U \subset H \subset U ^ {*} $ given by the injection $ i : U \rightarrow H $ is obtained by identifying $ H $ with its dual, taking the dual of $ i $, and endowing $ U ^ {*} $, the algebraic dual of $ U $, with the weak topology. For any real $ \lambda $, let $ \lambda H $ be the Hilbert space obtained from $ H $ by multiplying the norm on $ H $ by $ \lambda $.

The dual of the symmetric $ k $- fold tensor product $ S _ {k} ( U) $ is the space $ \mathop{\rm Pol} _ {k} ( U) $ of all homogeneous polynomials of degree $ k $ on $ U $. The value of $ F _ {k} \in \mathop{\rm Pol} _ {k} ( U) $ at $ u \in U $ is $ F _ {k} ( u ) = \langle F _ {k} , u ^ {\otimes k } \rangle _ {k! } $. Thus, for each $ k $ there is a triplet

$$ \tag{a1 } S _ {k} ( U) \subset \sqrt k! S _ {k} ( H) \subset \mathop{\rm Pol} _ {k} ( U) . $$

Taking the direct sum of the internal space $ S _ {k} ( U) $ and the Hilbert sum of the central spaces there results a triplet

$$ \tag{a2 } S( U) \subset \mathop{\rm Fock} ( H) \subset \mathop\widehat{ {\rm Pol}} ( U), $$

called dressed Fock space. The middle term is the usual Fock space

$$ \tag{a3 } \mathop{\rm Fock} ( H) = \oplus \sqrt k! S _ {k} ( H) . $$

The external space is the space $ \prod _ {k} \mathop{\rm Pol} _ {k} ( U) $ of all formal power series on $ U $. The value $ F( u ) $ at $ u \in U $ of such an $ F \in \mathop\widehat{ {\rm Pol}} ( U) $ is defined as $ \lim\limits _ \rightarrow \sum _ {k=} 1 ^ {N} F _ {k} ( u ) $, if this limit exists. For example, for any $ F = \sum F _ {k} \in \mathop{\rm Fock} ( H) $ one has

$$ \tag{a4 } F( u ) = \langle F, e ^ {u} \rangle , $$

where $ e ^ {u} = \sum ( k!) ^ {-} 1 u ^ {\otimes k } $.

A probabilized vector space is a structure

$$ \tag{a5 } ( U \dots X \supset \Omega , {\mathsf P} ) $$

where $ U $ and $ X $ are two spaces in duality and $ X = \mathop{\rm span} ( \Omega ) $ is linearly generated by the subset $ \Omega $ of $ X $. This subset is endowed with a Polish (or Suslin) topology such that any $ u \in U $ defines a Borel function $ u( \omega ) = \langle u , \omega \rangle $ on $ \Omega $. The space $ U $ contains a countable subset separating the points of $ \Omega $( so that the Borel $ \sigma $- field is generated by $ U $). Finally, $ {\mathsf P} $ is a probability measure on this $ \sigma $- field.

Assume, moreover, that the space of cylindrical polynomials $ P( \Omega ) = \mathop{\rm span} ( u ( \omega ) ^ {k} : u \in U, k = 0, 1, 2 , . . . ) $ is dense in $ L _ {2} ( \Omega ) $. Assume that the following bilinear form on $ U $ is a scalar product:

$$ \tag{a6 } b( u , v) = {\mathsf E} ( [ u ( \omega )- {\mathsf E} ( u ( \omega ))] [ v( \omega ) - {\mathsf E} ( v( \omega ))]) , $$

and let $ H $ be the completion of $ U $. For any $ k > 0 $, let $ \pi _ {k} $ denote the orthogonal projection of $ L _ {2} ( \Omega ) $ with range $ \overline{ {P _ {<} k ( \Omega ) }}\; $, the closure of $ \mathop{\rm span} ( u ( \omega ) ^ {j} : u \in U, j< k ) $. Let $ KO _ {k} $ be the orthogonal complement of $ \overline{ {P _ {<} k ( \Omega ) }}\; $ in $ \overline{ {P _ \leq k ( \Omega ) }}\; $. This space is called the $ k $- th homogeneous chaos. The space $ L _ {2} ( \Omega ) $ is the Hilbert direct sum of the $ KO _ {k} $. One says that $ L _ {2} ( \Omega ) $ admits a decomposition in chaos if for any $ k $ the following mapping is isometric:

$$ \sqrt k! S _ {k} ( H) \supset S _ {k} ( U) \ni \ Q \mapsto ^ { {I _ k} } Q - \pi _ {k} ( Q) \in \ KO _ {k} \subset L _ {2} ( \Omega ) . $$

The collection of these isometries for $ k = 0, 1 \dots $ is an isometry $ I $ whose inverse

$$ \tag{a7 } L _ {2} ( \Omega ) \rightarrow ^ { {I ^ {-}} 1 } \mathop{\rm Fock} ( H) ,\ \ f \rightarrow \widehat{f} , $$

extended to distributions on $ \Omega $, is the starting point of distribution calculus on $ \Omega $. Because of (a4), $ \widehat{f} $ is explicitly given by

$$ \tag{a8 } \widehat{f} ( u ) = \langle \widehat{f} , e ^ {u} \rangle = {\mathsf E} [ f \epsilon ^ {u} ] , $$

where $ \epsilon ^ {u} = I ^ {-} 1 ( e ^ {u} ) $.

Decomposition in chaos was discovered by N. Wiener (in the case $ \Omega $ is Wiener space), [a1]. Further contributions are due to Th.A. Dwyer and I. Segal ([a2], [a3]) and these have been important for constructive quantum field theory. K. Itô obtained a decomposition into chaos for Poisson probability spaces and interpreted $ I _ {k} ( f ) $ as iterated stochastic integrals. For formula (a8), extended to distributions for Gaussian probability spaces, cf. [a5], [a6], [a7], [a9], [a10]. There are links with Malliavin calculus, [a8].

For more material cf. e.g. also [a11], [a12]; Wick product and White noise analysis, and the references therein.

References

[a1] N. Wiener, "The homogeneous chaos" Amer. J. Math. , 60 (1938) pp. 897–936
[a2] Th.A., III Dwyer, "Partial differential equations in Fischer–Fock spaces for the Hilbert–Schmidt holomorphy type" Bull. Amer. Math. Soc. , 77 (1971) pp. 725–730
[a3] I. Segal, "Tensor algebras over Hilbert spaces, I" Trans. Amer. Math. Soc. , 81 (1956) pp. 106–134
[a4] K. Itô, "Multiple Wiener integral" J. Math. Soc. Japan (1951) pp. 157–169
[a5] P. Krée, "Solutions faibles d'equations aux dérivées fonctionelles II" , Sem. P. Lelong 1973/1974 , Lect. notes in math. , 474 , Springer (1974) pp. 16–47
[a6] P. Krée, R. Raczka, "Kernels and symbols of operators in quantum field theory" Ann. Inst. H. Poincaré (1978)
[a7] B. Lascar, "Propriétés locales des espaces de type Sobolev en dimension infinie" Comm. Partial Diff. Eq. , 1 : 6 (1976) pp. 561–584
[a8] D. Ocone, "Malliavin calculus and stochastic integral representation of functionals of diffusion processes" Stochastics , 12 (1984) pp. 161–185
[a9] M. Krée, "Propriété de trace en dimension infinie d'espaces du type Sobolev" C.R Acad. Sci. Paris , 279 (1974) pp. 157–160
[a10] M. Krée, "Propriété de trace en dimension infinie d'espaces de type Sobolev" Bull. Soc. Math. de France , 105 (1977) pp. 141–163
[a11] G. Kallianpur, "The role of reproducing kernel Hilbert spaces in the study of Gaussian processes" P. Ney (ed.) , Advances in probability and related topics , 2 , M. Dekker (1970) pp. 49–84
[a12] J. Neveu, "Processus aléatoires Gaussiens" , Univ. Montréal (1968)
How to Cite This Entry:
Wiener chaos decomposition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wiener_chaos_decomposition&oldid=14192
This article was adapted from an original article by P. Krée (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article