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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/w110060/w1100601.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w1100602.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w1100603.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w1100604.png" /> is a [[Probability space|probability space]]. The Wiener measure is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w1100605.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w1100606.png" />. Brownian motion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w1100607.png" /> is a [[Gaussian process|Gaussian process]] such that
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<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/w110060/w1100608.png" /></td> </tr></table>
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Given a Brownian motion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w1100609.png" />, one can form a new Brownian motion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006010.png" /> satisfying:
+
The [[Probability distribution|probability distribution]] of a [[Brownian motion|Brownian motion]]  $  B ( t, \omega ) $,
 +
$  t \geq  0 $,
 +
$  \omega \in \Omega $,
 +
where  $  ( \Omega, {\mathcal B}, {\mathsf P} ) $
 +
is a [[Probability space|probability space]]. The Wiener measure is denoted by  $  m $
 +
or  $  \mu  ^ {W} $.  
 +
Brownian motion $  B $
 +
is a [[Gaussian process|Gaussian process]] such that
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006011.png" /> is continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006012.png" /> for almost all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006013.png" />.
+
$$
 +
{\mathsf E} ( B ( t ) ) \equiv 0, \quad {\mathsf E} ( B ( t ) \cdot B ( s ) ) = \min  ( t, s ) .
 +
$$
  
ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006014.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006015.png" />.
+
Given a Brownian motion  $  B ( t, \omega ) $,
 +
one can form a new Brownian motion  $  {\overline{B}\; } ( t, \omega ) $
 +
satisfying:
  
Such a process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006016.png" /> is called a continuous version of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006017.png" />.
+
i)  $  {\overline{B}\; } ( t, \omega ) $
 +
is continuous in  $  t $
 +
for almost all  $  \omega $.
  
The Kolmogorov–Prokhorov theorem tells that the probability distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006018.png" /> of the Brownian motion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006019.png" /> can be introduced in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006020.png" /> of all continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006021.png" />.
+
ii)  $  {\mathsf P} ( {\overline{B}\; } ( t, \omega ) = B ( t, \omega ) ) = 1 $
 +
for every  $  t $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006022.png" /> be the topological Borel field (cf. also [[Borel field of sets|Borel field of sets]]) of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006023.png" />. The [[Measure space|measure space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006024.png" /> thus obtained is the Wiener measure space.
+
Such a process  $  {\overline{B}\; } ( t, \omega ) $
 +
is called a continuous version of $  B ( t, \omega ) $.
  
The integral of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006025.png" />-measurable functional on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006026.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006027.png" /> is defined in the usual manner. (See also [[Stochastic integral|Stochastic integral]].)
+
The Kolmogorov–Prokhorov theorem tells that the probability distribution  $  m $
 +
of the Brownian motion  $  B ( t ) $
 +
can be introduced in the space  $  C = C [ 0, \infty ) $
 +
of all continuous functions on $  [ 0, \infty ) $.
  
An elementary and important example of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006028.png" />-measurable functional of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006029.png" /> is a stochastic bilinear form, given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006030.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006031.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006032.png" />-function. It is usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006033.png" />. It is, in fact, defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006034.png" /> for smooth functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006035.png" />. For a general <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006037.png" /> can be approximated by stochastic bilinear forms defined by smooth functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006038.png" />. An integral of this type is called a Wiener integral. Under certain restrictions, such as non-anticipation, the integral can be extended to the case where the integrand is a functional of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006040.png" />. And an even more general case has been proposed.
+
Let  $  {\mathcal B} $
 +
be the topological Borel field (cf. also [[Borel field of sets|Borel field of sets]]) of subsets of $  C $.  
 +
The [[Measure space|measure space]]  $  ( C, {\mathcal B}, m ) $
 +
thus obtained is the Wiener measure space.
  
The class of general (non-linear) functionals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006041.png" /> is introduced as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006042.png" /> be the [[Hilbert space|Hilbert space]] of all complex-valued, square-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006043.png" />-integrable functionals on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006044.png" />. Then, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006045.png" /> admits a direct sum decomposition ([[Fock space|Fock space]])
+
The integral of a  $  {\mathcal B} $-
 +
measurable functional on  $  C $
 +
with respect to  $  m $
 +
is defined in the usual manner. (See also [[Stochastic integral|Stochastic integral]].)
  
<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/w110060/w11006046.png" /></td> </tr></table>
+
An elementary and important example of a  $  {\mathcal B} $-
 +
measurable functional of  $  y \in C $
 +
is a stochastic bilinear form, given by  $  \langle  { {\dot{y} } , f } \rangle $,
 +
where  $  f $
 +
is an  $  L _ {2} [ 0, \infty ) $-
 +
function. It is usually denoted by  $  f ( y ) $.
 +
It is, in fact, defined by  $  - \int _ {0}  ^  \infty  {y ( t ) {\dot{f} } ( t ) }  {dt } $
 +
for smooth functions  $  f $.
 +
For a general  $  f $,
 +
$  f ( y ) $
 +
can be approximated by stochastic bilinear forms defined by smooth functions  $  f $.
 +
An integral of this type is called a Wiener integral. Under certain restrictions, such as non-anticipation, the integral can be extended to the case where the integrand is a functional of  $  t $
 +
and  $  y $.  
 +
And an even more general case has been proposed.
  
The subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006047.png" /> is spanned by the Fourier–Hermite polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006048.png" />, which are of the form
+
The class of general (non-linear) functionals of  $  y $
 +
is introduced as follows. Let  $  H $
 +
be the [[Hilbert space|Hilbert space]] of all complex-valued, square- $  m $-
 +
integrable functionals on  $  C $.  
 +
Then, $  H $
 +
admits a direct sum decomposition ([[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/w110/w110060/w11006049.png" /></td> </tr></table>
+
$$
 +
H = \oplus _ { n } {\mathcal H} _ {n} .
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006051.png" /> is a complete [[Orthonormal system|orthonormal system]] in the Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006052.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006053.png" /> can be interpreted as the space of multiple Wiener integrals of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006054.png" />, due to K. Itô.
+
The subspace  $  {\mathcal H} _ {n} $
 +
is spanned by the Fourier–Hermite polynomials of degree  $  n $,
 +
which are of the form
 +
 
 +
$$
 +
\prod _ { j } H _ {n _ {j}  } \left ( {
 +
\frac{\left \langle  {y,f _ {j} } \right \rangle }{\sqrt 2 }
 +
} \right ) ,
 +
$$
 +
 
 +
where $  \Sigma n _ {j} = n $
 +
and $  \{ f _ {j} \} $
 +
is a complete [[Orthonormal system|orthonormal system]] in the Hilbert space $  L _ {2} [ 0, \infty ) $.  
 +
The space $  H _ {n} $
 +
can be interpreted as the space of multiple Wiener integrals of degree $  n $,  
 +
due to K. Itô.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Cameron,  W.T. Martin,  "The orthogonal development of non-linear functionals in series of Fourier–Hermite functionals"  ''Ann. of Math. (2)'' , '''48'''  pp. 385–392</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T. Hida,  "Brownian motion" , ''Applications of Mathematics'' , '''11''' , Springer  (1980)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Cameron,  W.T. Martin,  "The orthogonal development of non-linear functionals in series of Fourier–Hermite functionals"  ''Ann. of Math. (2)'' , '''48'''  pp. 385–392</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T. Hida,  "Brownian motion" , ''Applications of Mathematics'' , '''11''' , Springer  (1980)</TD></TR></table>

Latest revision as of 08:29, 6 June 2020


The probability distribution of a Brownian motion $ B ( t, \omega ) $, $ t \geq 0 $, $ \omega \in \Omega $, where $ ( \Omega, {\mathcal B}, {\mathsf P} ) $ is a probability space. The Wiener measure is denoted by $ m $ or $ \mu ^ {W} $. Brownian motion $ B $ is a Gaussian process such that

$$ {\mathsf E} ( B ( t ) ) \equiv 0, \quad {\mathsf E} ( B ( t ) \cdot B ( s ) ) = \min ( t, s ) . $$

Given a Brownian motion $ B ( t, \omega ) $, one can form a new Brownian motion $ {\overline{B}\; } ( t, \omega ) $ satisfying:

i) $ {\overline{B}\; } ( t, \omega ) $ is continuous in $ t $ for almost all $ \omega $.

ii) $ {\mathsf P} ( {\overline{B}\; } ( t, \omega ) = B ( t, \omega ) ) = 1 $ for every $ t $.

Such a process $ {\overline{B}\; } ( t, \omega ) $ is called a continuous version of $ B ( t, \omega ) $.

The Kolmogorov–Prokhorov theorem tells that the probability distribution $ m $ of the Brownian motion $ B ( t ) $ can be introduced in the space $ C = C [ 0, \infty ) $ of all continuous functions on $ [ 0, \infty ) $.

Let $ {\mathcal B} $ be the topological Borel field (cf. also Borel field of sets) of subsets of $ C $. The measure space $ ( C, {\mathcal B}, m ) $ thus obtained is the Wiener measure space.

The integral of a $ {\mathcal B} $- measurable functional on $ C $ with respect to $ m $ is defined in the usual manner. (See also Stochastic integral.)

An elementary and important example of a $ {\mathcal B} $- measurable functional of $ y \in C $ is a stochastic bilinear form, given by $ \langle { {\dot{y} } , f } \rangle $, where $ f $ is an $ L _ {2} [ 0, \infty ) $- function. It is usually denoted by $ f ( y ) $. It is, in fact, defined by $ - \int _ {0} ^ \infty {y ( t ) {\dot{f} } ( t ) } {dt } $ for smooth functions $ f $. For a general $ f $, $ f ( y ) $ can be approximated by stochastic bilinear forms defined by smooth functions $ f $. An integral of this type is called a Wiener integral. Under certain restrictions, such as non-anticipation, the integral can be extended to the case where the integrand is a functional of $ t $ and $ y $. And an even more general case has been proposed.

The class of general (non-linear) functionals of $ y $ is introduced as follows. Let $ H $ be the Hilbert space of all complex-valued, square- $ m $- integrable functionals on $ C $. Then, $ H $ admits a direct sum decomposition (Fock space)

$$ H = \oplus _ { n } {\mathcal H} _ {n} . $$

The subspace $ {\mathcal H} _ {n} $ is spanned by the Fourier–Hermite polynomials of degree $ n $, which are of the form

$$ \prod _ { j } H _ {n _ {j} } \left ( { \frac{\left \langle {y,f _ {j} } \right \rangle }{\sqrt 2 } } \right ) , $$

where $ \Sigma n _ {j} = n $ and $ \{ f _ {j} \} $ is a complete orthonormal system in the Hilbert space $ L _ {2} [ 0, \infty ) $. The space $ H _ {n} $ can be interpreted as the space of multiple Wiener integrals of degree $ n $, due to K. Itô.

References

[a1] R. Cameron, W.T. Martin, "The orthogonal development of non-linear functionals in series of Fourier–Hermite functionals" Ann. of Math. (2) , 48 pp. 385–392
[a2] T. Hida, "Brownian motion" , Applications of Mathematics , 11 , Springer (1980)
How to Cite This Entry:
Wiener measure(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wiener_measure(2)&oldid=19019
This article was adapted from an original article by T. Hida (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article