View source for Wiener measure

<!--
w0979301.png
$#A+1 = 20 n = 0
$#C+1 = 20 : ~/encyclopedia/old_files/data/W097/W.0907930 Wiener measure
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

The [[Probability measure|probability measure]]  $  \mu _ {W} $
on the space  $  C[ 0, 1] $
of continuous real-valued functions  $  x $
on the interval  $  [ 0, 1] $, 
defined as follows. Let  $  0 < t _ {1} < \dots < t _ {n} \leq  1 $
be an arbitrary sample of points from  $  [ 0, 1] $
and let  $  A _ {1} \dots A _ {n} $
be Borel sets on the real line. Let  $  C( t _ {1} \dots t _ {n} ;  A _ {1} \dots A _ {n} ) $
denote the set of functions  $  x \in C[ 0, 1] $
for which  $  x( t _ {k} ) \in A _ {k} $, 
$  k = 1 \dots n $. 
Then

$$ \tag{* }
\mu _ {W} ( C ( t _ {1} \dots t _ {n} ;  A _ {1} \dots A _ {n} )) =
$$

$$ 
= \ 
\int\limits _ { A _ 1 } p ( t _ {1} , x _ {1} )  dx _ {1}  \int\limits _ { A _ 2 } p ( t _ {2} - t _ {1} , x _ {2} - x _ {1} )  dx _ {2} \dots
$$

$$ 
{} \dots \int\limits _ { A _ n } p ( t _ {n} - t _ {n-} 1 , x _ {n} - x _ {n-} 1 )  dx _ {n} ,
$$

where

$$ 
p ( t, x)  =  {
\frac{1}{\sqrt {2 \pi t } }
 } e ^ {- x  ^ {2} / 2 t } .
$$

Using the theorem on the extension of a measure it is possible to define the value of the measure  $  \mu _ {W} $
on all Borel sets of  $  C[ 0, 1] $
on the basis of equation (*).

====Comments====
The Wiener measure was introduced by N. Wiener [[#References|[a1]]] in 1923; it was the first major extension of integration theory beyond a finite-dimensional setting. The construction outlined above extends easily to define Wiener measure  $  \mu _ {W} $
on  $  C [ 0, \infty ) $. 
The coordinate process  $  x( t) $
is then known as [[Brownian motion|Brownian motion]] or the [[Wiener process|Wiener process]]. Its formal derivative  "dxt/dt"  is known as Gaussian [[White noise|white noise]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Wiener,   "Differential space"  ''J. Math. &amp; Phys.'' , '''2'''  (1923)  pp. 132–174</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T. Hida,   "Brownian motion" , Springer  (1980)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  I. Karatzas,   S.E. Shreve,   "Brownian motion and stochastic calculus" , Springer  (1988)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  L. Partzsch,   "Vorlesungen zum eindimensionalen Wienerschen Prozess" , Teubner  (1984)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J. Yeh,   "Stochastic processes and the Wiener integral" , M. Dekker  (1973)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  S. Albeverio,   J.E. Fenstad,   R. Høegh-Krohn,   T. Lindstrøm,   "Nonstandard methods in stochastic analysis and mathematical physics" , Acad. Press  (1986)</TD></TR></table>