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{{MSC|60J65}}
 
{{MSC|60J65}}
  
 
[[Category:Markov processes]]
 
[[Category:Markov processes]]
  
A homogeneous [[Gaussian process|Gaussian process]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w0979401.png" /> with independent increments. A Wiener process serves as one of the models of [[Brownian motion|Brownian motion]]. A simple transformation will convert a Wiener process into the "standard" Wiener process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w0979402.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w0979403.png" />, for which
+
A homogeneous [[Gaussian process|Gaussian process]] $  X( t) $
 +
with independent increments. A Wiener process serves as one of the models of [[Brownian motion|Brownian motion]]. A simple transformation will convert a Wiener process into the "standard" Wiener process $  X( t) $,  
 +
$  t \geq  0 $,  
 +
for which
  
<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/w097940/w0979404.png" /></td> </tr></table>
+
$$
 +
X ( 0)  = 0,\ \
 +
{\mathsf E} ( X ( t) - X ( s))  = 0,
 +
$$
  
<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/w097940/w0979405.png" /></td> </tr></table>
+
$$
 +
{\mathsf D} [ X ( t) - X ( s)]  = t - s,\  s \leq  t.
 +
$$
  
 
For these average values and incremental variances, this is the only almost-surely continuous process with independent increments. In what follows, the Wiener process will be understood to be this process.
 
For these average values and incremental variances, this is the only almost-surely continuous process with independent increments. In what follows, the Wiener process will be understood to be this process.
  
The Wiener process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w0979406.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w0979407.png" />, can also be defined as the Gaussian process with zero expectation and covariance function
+
The Wiener process $  X( t) $,  
 +
0 \leq  t \leq  1 $,  
 +
can also be defined as the Gaussian process with zero expectation and covariance function
 +
 
 +
$$
 +
B ( s, t)  =  \min  ( s, t).
 +
$$
 +
 
 +
The Wiener process  $  X= X( t) $,
 +
$  t \geq  0 $,
 +
may also be defined as the homogeneous [[Markov process|Markov process]] with transition function
  
<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/w097940/w0979408.png" /></td> </tr></table>
+
$$
 +
P ( t, x, \Gamma )  = \int\limits _  \Gamma  p ( t, x, y)  dy,
 +
$$
  
The Wiener process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w0979409.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794010.png" />, may also be defined as the homogeneous [[Markov process|Markov process]] with transition function
+
where the transition density  $  p( t, x, y) $
 +
is the fundamental solution of the parabolic differential equation
  
<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/w097940/w09794011.png" /></td> </tr></table>
+
$$
  
where the transition density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794012.png" /> is the fundamental solution of the parabolic differential equation
+
\frac{\partial  p }{\partial  t }
 +
  = \
 +
{
 +
\frac{1}{2}
 +
}
  
<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/w097940/w09794013.png" /></td> </tr></table>
+
\frac{\partial  ^ {2} p }{\partial  x  ^ {2} }
 +
,
 +
$$
  
 
given by the formula
 
given by the formula
  
<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/w097940/w09794014.png" /></td> </tr></table>
+
$$
 +
p ( t, x, y)  = \
 +
 
 +
\frac{1}{\sqrt {2 \pi t } }
 +
 
 +
e ^ {-( y- x)  ^ {2} / 2t } .
 +
$$
  
The transition function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794015.png" /> is invariant with respect to translations in the phase space:
+
The transition function $  P( t, x, \Gamma ) $
 +
is invariant with respect to translations in the phase 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/w097940/w09794016.png" /></td> </tr></table>
+
$$
 +
P ( t, x + y, \Gamma )  = P ( t, x, \Gamma - y),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794017.png" /> denotes the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794018.png" />.
+
where $  \Gamma - y $
 +
denotes the set $  \{ {z } : {z + y \in \Gamma } \} $.
  
The Wiener process is the continuous analogue of the [[Random walk|random walk]] of a particle which, at discrete moments of time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794019.png" /> (multiples of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794020.png" />), is randomly displaced by a quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794021.png" />, independent of the past (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794023.png" />); more precisely, if
+
The Wiener process is the continuous analogue of the [[Random walk|random walk]] of a particle which, at discrete moments of time $  t= k \Delta t $(
 +
multiples of $  \Delta t $),  
 +
is randomly displaced by a quantity $  \Delta X( t) $,  
 +
independent of the past ( $  {\mathsf E} \Delta X( t) = 0 $,  
 +
$  {\mathsf D} \Delta X( t) = \Delta t $);  
 +
more precisely, if
  
<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/w097940/w09794024.png" /></td> </tr></table>
+
$$
 +
X ( t)  = \sum _{k = 0 } ^ {m-1}
 +
\Delta X \left ( {
 +
\frac{k}{n}
 +
} \right ) + ( nt - m)
 +
\Delta X \left ( {
 +
\frac{m}{n}
 +
} \right ) ,\ \
 +
0 \leq  t \leq  1,
 +
$$
  
is the random trajectory of the motion of such a particle on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794025.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794026.png" /> is the integer part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794028.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794030.png" /> is the corresponding probability distribution in the space of continuous functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794032.png" />), then the probability distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794033.png" /> of the trajectory of the Wiener process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794035.png" />, is the limit (in the sense of weak convergence) of the distributions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794036.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794037.png" />.
+
is the random trajectory of the motion of such a particle on the interval $  [ 0, 1] $(
 +
where $  m= [ nt] $
 +
is the integer part of $  nt $,  
 +
$  X( t) = nt \Delta X ( 0) $
 +
if $  0 \leq  t < 1/n $
 +
and $  {\mathsf P} _ {n} $
 +
is the corresponding probability distribution in the space of continuous functions $  x= x( t) $,  
 +
0 \leq  t \leq  1 $),  
 +
then the probability distribution $  {\mathsf P} $
 +
of the trajectory of the Wiener process $  X( t) $,  
 +
0 \leq  t \leq  1 $,  
 +
is the limit (in the sense of weak convergence) of the distributions $  {\mathsf P} _ {n} $
 +
as $  n \rightarrow \infty $.
  
As a function with values in the Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794038.png" /> of all random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794039.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794040.png" />, in which the scalar product is defined by the formula
+
As a function with values in the Hilbert space $  L _ {2} ( \Omega ) $
 +
of all random variables $  X $
 +
with $  {\mathsf E} X  ^ {2} < \infty $,  
 +
in which the scalar product is defined by the formula
  
<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/w097940/w09794041.png" /></td> </tr></table>
+
$$
 +
\langle  X _ {1} , X _ {2} \rangle  = {\mathsf E} X _ {1} X _ {2} ,
 +
$$
  
the Wiener process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794043.png" />, may be canonically represented as follows:
+
the Wiener process $  X = X( t) $,  
 +
0 \leq  t \leq  1 $,  
 +
may be canonically represented as follows:
  
<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/w097940/w09794044.png" /></td> </tr></table>
+
$$
 +
X ( t)  = \sum _ {k = 0 } ^  \infty  z _ {k} \phi _ {k} ( t),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794045.png" /> are independent Gaussian variables:
+
where $  z _ {k} $
 +
are independent Gaussian variables:
  
<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/w097940/w09794046.png" /></td> </tr></table>
+
$$
 +
{\mathsf E} z _ {k}  = 0,\ \
 +
{\mathsf D} z _ {k}  =
 +
\frac{1}{\left [ {
 +
\frac \pi {2}
 +
} ( 2k + 1) \right ]  ^ {2} }
 +
,
 +
$$
  
 
and
 
and
  
<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/w097940/w09794047.png" /></td> </tr></table>
+
$$
 +
\phi _ {k} ( t)  = \
 +
\sin \left [ {
 +
\frac \pi {2}
 +
} ( 2k + 1) t \right ] ,\ \
 +
k = 0, 1 \dots
 +
$$
  
are the eigenfunctions of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794048.png" /> defined by the formula
+
are the eigenfunctions of the operator $  B $
 +
defined by the formula
  
<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/w097940/w09794049.png" /></td> </tr></table>
+
$$
 +
B \phi ( t)  = \int\limits _ { 0 } ^ { 1 }  B ( s, t) \phi ( s)  ds
 +
$$
  
in the Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794050.png" /> of all square-integrable (with respect to Lebesgue measure) functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794051.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794052.png" />.
+
in the Hilbert space $  L _ {2} [ 0, 1] $
 +
of all square-integrable (with respect to Lebesgue measure) functions $  \phi $
 +
on $  [ 0, 1] $.
  
 
Almost-all trajectories of the Wiener process have the following properties:
 
Almost-all trajectories of the Wiener process have the following properties:
  
<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/w097940/w09794053.png" /></td> </tr></table>
+
$$
 +
{\lim\limits  \sup } _ {h \rightarrow \infty } \
 +
 
 +
\frac{X ( h) }{\sqrt {2 h  \mathop{\rm ln}  \mathop{\rm ln}  {
 +
\frac{1}{h}
 +
} } }
 +
  = 1,\ \
 +
X ( 0)  = 0,
 +
$$
  
 
which is the law of the iterated logarithm;
 
which is the law of the iterated logarithm;
  
<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/w097940/w09794054.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {h \rightarrow + 0 } \
 +
\sup _ {0 \leq  t \leq  \delta - h } \
  
characterizing the modulus of continuity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794055.png" />; and
+
\frac{| X ( t+ h) - X ( t) | }{\sqrt {2 h  \mathop{\rm ln}  {
 +
\frac \delta {h}
 +
} } }
 +
  =  1,
 +
$$
 +
 
 +
characterizing the modulus of continuity on $  [ 0, h] $;  
 +
and
  
<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/w097940/w09794056.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {h \rightarrow \infty }  \sum_{k = 0 } ^ {n-1} | \Delta X ( kh) |  ^ {2}  = \delta ,
 +
$$
  
<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/w097940/w09794057.png" /></td> </tr></table>
+
$$
 +
=
 +
\frac \delta {n}
 +
,\  \Delta X ( t)  = X ( t+ h) - X ( t).
 +
$$
  
When applied to the Wiener process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794059.png" />, the law of the iterated logarithm reads:
+
When applied to the Wiener process $  X _ {1} ( t) = tX( 1/t) $,  
 +
$  0 \leq  t < \infty $,  
 +
the law of the iterated logarithm reads:
  
<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/w097940/w09794060.png" /></td> </tr></table>
+
$$
 +
{\lim\limits  \sup } _ {t \rightarrow \infty } 
 +
\frac{X ( t) }{\sqrt {2 t  \mathop{\rm ln}  \mathop{\rm ln}  t } }
 +
  = 1.
 +
$$
  
The distributions of the maximum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794061.png" />, of the time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794062.png" /> at which the Brownian particle first reaches a fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794063.png" /> and of the location <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794064.png" /> of the maximum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794065.png" /> give insight in the nature of the movement of a Brownian particle; these distributions are given by the following formulas:
+
The distributions of the maximum $  \max _ {0 \leq  s \leq  t }  X( s) $,  
 +
of the time $  r _ {x} $
 +
at which the Brownian particle first reaches a fixed point $  x > 0 $
 +
and of the location $  \tau $
 +
of the maximum $  \max _ {0 \leq  s \leq  t }  X( s) $
 +
give insight in the nature of the movement of a Brownian particle; these distributions are given by the following formulas:
  
<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/w097940/w09794066.png" /></td> </tr></table>
+
$$
 +
{\mathsf P} \left \{ \max _ {0 \leq  s \leq  t }  X( s) \right \}  =
 +
\frac{2}{\sqrt {2 \pi t } }
 +
\int\limits _ { x } ^  \infty  e ^ {- u  ^ {2} /2t }  du ,
 +
$$
  
<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/w097940/w09794067.png" /></td> </tr></table>
+
$$
 +
P \{ \tau _ {x} \geq  t \}  = P \left \{ \max
 +
_ {0 \leq  s \leq  t }  X ( s) \leq  x \right \} =
 +
$$
  
<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/w097940/w09794068.png" /></td> </tr></table>
+
$$
 +
= \
 +
\sqrt {
 +
\frac{2} \pi
 +
}  \int\limits _ { 0 } ^ { {x/ }  \sqrt t } e ^ {- u  ^ {2} /2 }  d u ,
 +
$$
  
while the simultaneous density of the maximum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794069.png" /> and its location <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794070.png" /> is given by:
+
while the simultaneous density of the maximum $  \max _ {0 \leq  s \leq  t }  X( s) $
 +
and its location $  \tau $
 +
is 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/w097940/w09794071.png" /></td> </tr></table>
+
$$
 +
{\mathsf P} \{ \tau _ {x} \geq  t \}  = \
 +
\sqrt {
 +
\frac{2} \pi
 +
}  \int\limits _ { 0 } ^ { {x }  / \sqrt t }
 +
e ^ {- u  ^ {2} /2t }  d u ,
 +
$$
  
<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/w097940/w09794072.png" /></td> </tr></table>
+
$$
 +
< < t ,\  0  \leq  x  < \infty .
 +
$$
  
(The laws of the Wiener process remain unchanged on transforming the phase space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794073.png" />.) The joint distribution of the maximum point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794074.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794075.png" />, and of the maximum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794076.png" /> itself has the probability density
+
(The laws of the Wiener process remain unchanged on transforming the phase space $  x \rightarrow - x $.)  
 +
The joint distribution of the maximum point $  \tau $,  
 +
0 \leq  \tau \leq  t $,  
 +
and of the maximum $  \max _ {0\leq  s \leq  t }  X( s) $
 +
itself has the probability density
  
<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/w097940/w09794077.png" /></td> </tr></table>
+
$$
 +
p ( s, x)  = \
 +
{
 +
\frac{1}{\pi \sqrt {s ( t- s) } }
 +
}
 +
{
 +
\frac{x}{s}
 +
}
 +
e ^ {- x  ^ {2} / 2s } ,
 +
$$
  
<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/w097940/w09794078.png" /></td> </tr></table>
+
$$
 +
0  \leq  s  \leq  t,\  0 \leq  x < \infty ,
 +
$$
  
while the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794079.png" /> by itself (with probability one there is only one maximum on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794080.png" />) is distributed according to the [[Arcsine law|arcsine law]]:
+
while the point $  \tau $
 +
by itself (with probability one there is only one maximum on the interval 0 \leq  s \leq  t $)  
 +
is distributed according to the [[Arcsine law|arcsine law]]:
  
<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/w097940/w09794081.png" /></td> </tr></table>
+
$$
 +
{\mathsf P} \{ \tau \leq  s \}  = \
 +
{
 +
\frac{2} \pi
 +
}  \mathop{\rm arc}  \sin  \sqrt {
 +
\frac{s}{t}
 +
} ,\ \
 +
0 \leq  s \leq  t,
 +
$$
  
 
with the probability density:
 
with the probability density:
  
<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/w097940/w09794082.png" /></td> </tr></table>
+
$$
 +
p ( s)  = {
 +
\frac{1}{\pi \sqrt s( t- s) }
 +
} ,\ \
 +
0 \leq  s \leq  t.
 +
$$
 +
 
 +
The following properties of the Wiener process are readily deduced from the formulas given above. The Brownian trajectory is nowhere differentiable; on starting from any point  $  x $
 +
this trajectory crosses the "level" $  x $(
 +
returns to its initial point) infinitely many times with probability one, however short the time  $  \delta $;  
 +
the Brownian trajectory passes through all points  $  x $(
 +
more precisely,  $  \tau _ {x} < \infty $)
 +
with probability one (the most probable value of  $  \tau _ {x} $
 +
is of the order  $  x  ^ {2} $
 +
for large  $  x $);  
 +
this trajectory, if considered on a fixed interval  $  [ 0, t] $,
 +
tends to attain the extremal values near the end-points  $  s= 0 $
 +
and  $  s= t $.
  
The following properties of the Wiener process are readily deduced from the formulas given above. The Brownian trajectory is nowhere differentiable; on starting from any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794083.png" /> this trajectory crosses the "level" <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794084.png" /> (returns to its initial point) infinitely many times with probability one, however short the time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794085.png" />; the Brownian trajectory passes through all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794086.png" /> (more precisely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794087.png" />) with probability one (the most probable value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794088.png" /> is of the order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794089.png" /> for large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794090.png" />); this trajectory, if considered on a fixed interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794091.png" />, tends to attain the extremal values near the end-points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794092.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794093.png" />.
+
Since a Wiener process is a homogeneous Markov process, there exists an invariant measure  $  Q( d x) $
 +
for it, namely:
  
Since a Wiener process is a homogeneous Markov process, there exists an invariant measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794094.png" /> for it, namely:
+
$$
 +
Q ( A)  \equiv  \int\limits Q ( dx) P ( t, x, A),
 +
$$
  
<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/w097940/w09794095.png" /></td> </tr></table>
+
which, since the transition function  $  P( t, x, A) $
 +
has been seen to be invariant, coincides with the Lebesgue measure on the real line: $  Q( d x) = d x $.
 +
The time  $  T( A) $
 +
which a Brownian particle spends in  $  A $
 +
between the times 0 and  $  T $
 +
is such that
  
which, since the transition function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794096.png" /> has been seen to be invariant, coincides with the Lebesgue measure on the real line: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794097.png" />. The time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794098.png" /> which a Brownian particle spends in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w09794099.png" /> between the times 0 and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w097940100.png" /> is 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/w097/w097940/w097940101.png" /></td> </tr></table>
+
\frac{T ( A _ {1} ) }{T ( A _ {2} ) }
 +
  \rightarrow \
  
as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w097940102.png" />, with probability one for any bounded Borel sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w097940103.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w097940104.png" />.
+
\frac{Q ( A _ {1} ) }{Q ( A _ {2} ) }
  
Wiener random fields, introduced by P. Lévy [[#References|[3]]], are analogues of the Wiener process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w097940105.png" /> for a vector parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w097940106.png" />.
+
$$
  
====References====
+
as  $  T \rightarrow \infty $,
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> K. Itô, H.P. McKean jr., "Diffusion processes and their sample paths" , Springer (1974) {{MR|0345224}} {{ZBL|0285.60063}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian) {{MR|0251754}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> P. Lévy, "Processus stochastiques et mouvement Brownien" , Gauthier-Villars (1965) {{MR|0190953}} {{ZBL|0137.11602}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> V.P. Pavlov, "Brownian motion" , ''Large Soviet Encyclopaedia'' , '''4''' (In Russian)</TD></TR></table>
+
with probability one for any bounded Borel sets  $  A _ {1} $
 +
and $  A _ {2} $.
  
 +
Wiener random fields, introduced by P. Lévy {{Cite|L}}, are analogues of the Wiener process  $  X = X( t) $
 +
for a vector parameter  $  t = ( t _ {1} \dots t _ {n} ) $.
  
 +
====References====
 +
{|
 +
|valign="top"|{{Ref|IM}}|| K. Itô, H.P. McKean jr., "Diffusion processes and their sample paths" , Springer (1974) {{MR|0345224}} {{ZBL|0285.60063}}
 +
|-
 +
|valign="top"|{{Ref|PR}}|| Yu.V. Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian) {{MR|0251754}} {{ZBL|}}
 +
|-
 +
|valign="top"|{{Ref|L}}|| P. Lévy, "Processus stochastiques et mouvement Brownien" , Gauthier-Villars (1965) {{MR|0190953}} {{ZBL|0137.11602}}
 +
|-
 +
|valign="top"|{{Ref|P}}|| V.P. Pavlov, "Brownian motion" , ''Large Soviet Encyclopaedia'' , '''4''' (In Russian)
 +
|}
  
 
====Comments====
 
====Comments====
The Wiener process is more commonly referred to as [[Brownian motion|Brownian motion]] in the Western literature. It is by far the most important construct in stochastic analysis. See [[#References|[a1]]][[#References|[a3]]] for up-to-date accounts of its properties. Of particular importance is the theory of local time. The occupation time of a Borel set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w097940107.png" /> on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w097940108.png" /> is:
+
The Wiener process is more commonly referred to as [[Brownian motion|Brownian motion]] in the Western literature. It is by far the most important construct in stochastic analysis. See {{Cite|Du}}{{Cite|RY}} for up-to-date accounts of its properties. Of particular importance is the theory of local time. The occupation time of a Borel set $  B \subset  \mathbf R $
 +
on the interval $  [ 0, t] $
 +
is:
  
<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/w097940/w097940109.png" /></td> </tr></table>
+
$$
 +
\Gamma _ {t} ( B)  = \int\limits _ { 0 } ^ { t }  I _ {B} ( X( s))  ds .
 +
$$
  
There exists an almost-surely jointly-continuous [[Random field|random field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w097940110.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w097940111.png" /> such that
+
There exists an almost-surely jointly-continuous [[Random field|random field]] $  L ( t, x) $
 +
for $  ( t, x) \in \mathbf R _ {+} \times \mathbf R $
 +
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/w097/w097940/w097940112.png" /></td> </tr></table>
+
$$
 +
\Gamma _ {t} ( B)  = 2 \int\limits _ { B } L( t, x)  dx;
 +
$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w097940113.png" /> is the local time at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w097940114.png" />. For fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w097940115.png" />, sample paths of the process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w097940116.png" /> are increasing and continuous but singular with respect to Lebesgue measure.
+
$  L( t, x) $
 +
is the local time at $  x $.  
 +
For fixed $  x \in \mathbf R $,  
 +
sample paths of the process $  t \mapsto L( t, x) $
 +
are increasing and continuous but singular with respect to Lebesgue measure.
  
 
See also [[Markov process|Markov process]]; [[Stochastic differential equation|Stochastic differential equation]].
 
See also [[Markov process|Markov process]]; [[Stochastic differential equation|Stochastic differential equation]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Durrett, "Brownian motion and martingales in analysis" , Wadsworth (1984) {{MR|0750829}} {{ZBL|0554.60075}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> I. Karatzas, S.E. Shreve, "Brownian motion and stochastic calculus" , Springer (1988) {{MR|0917065}} {{ZBL|0638.60065}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> D. Revuz, M. Yor, "Continuous martingales and Brownian motion" , Springer (1990) {{MR|1725357}} {{MR|1303781}} {{MR|1083357}} {{ZBL|1087.60040}} {{ZBL|0917.60006}} {{ZBL|0804.60001}} {{ZBL|0731.60002}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> E.B. Dynkin, "Markov processes" , '''1''' , Springer (1965) (Translated from Russian) {{MR|0193671}} {{ZBL|0132.37901}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> W. Feller, "An introduction to probability theory and its applications" , '''1–2''' , Wiley (1968–1971) {{MR|0779091}} {{MR|0779090}} {{MR|0270403}} {{MR|0228020}} {{MR|1534302}} {{MR|0243559}} {{MR|0242202}} {{MR|0210154}} {{MR|1570945}} {{MR|0088081}} {{MR|1528130}} {{MR|0067380}} {{MR|0038583}} {{ZBL|0598.60003}} {{ZBL|0598.60002}} {{ZBL|0219.60003}} {{ZBL|0155.23101}} {{ZBL|0158.34902}} {{ZBL|0151.22403}} {{ZBL|0138.10207}} {{ZBL|0115.35308}} {{ZBL|0077.12201}} {{ZBL|0039.13201}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> I.I. [I.I. Gikhman] Gihman, A.V. [A.V. Skorokhod] Skorohod, "The theory of stochastic processes" , '''III''' , Springer (1975) (Translated from Russian) {{MR|0375463}} {{ZBL|0305.60027}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> T. Hida, "Brownian motion" , Springer (1980) {{MR|0566333}} {{MR|0562914}} {{ZBL|0432.60002}} {{ZBL|0423.60063}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> F. Spitzer, "Principles of random walk" , v. Nostrand (1964) {{MR|0171290}} {{ZBL|0119.34304}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> J. Yeh, "Stochastic processes and the Wiener integral" , M. Dekker (1973) {{MR|0474528}} {{ZBL|0277.60018}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> J.L. Doob, "Classical potential theory and its probabilistic counterpart" , Springer (1984) {{MR|0731258}} {{ZBL|0549.31001}} </TD></TR></table>
+
{|
 +
|valign="top"|{{Ref|Du}}|| R. Durrett, "Brownian motion and martingales in analysis", Wadsworth (1984) {{MR|0750829}} {{ZBL|0554.60075}}
 +
|-
 +
|valign="top"|{{Ref|KS}}|| I. Karatzas, S.E. Shreve, "Brownian motion and stochastic calculus", Springer (1988) {{MR|0917065}} {{ZBL|0638.60065}}
 +
|-
 +
|valign="top"|{{Ref|RY}}|| D. Revuz, M. Yor, "Continuous martingales and Brownian motion", Springer (1990) {{MR|1725357}} {{MR|1303781}} {{MR|1083357}} {{ZBL|1087.60040}} {{ZBL|0917.60006}} {{ZBL|0804.60001}} {{ZBL|0731.60002}}
 +
|-
 +
|valign="top"|{{Ref|Dy}}|| E.B. Dynkin, "Markov processes", '''1''', Springer (1965) (Translated from Russian) {{MR|0193671}} {{ZBL|0132.37901}}
 +
|-
 +
|valign="top"|{{Ref|F}}|| W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its applications"]], '''1–2''', Wiley (1968–1971)
 +
|-
 +
|valign="top"|{{Ref|GS}}|| I.I. Gihman, A.V. Skorohod, "The theory of stochastic processes", '''III''', Springer (1975) (Translated from Russian) {{MR|0375463}} {{ZBL|0305.60027}}
 +
|-
 +
|valign="top"|{{Ref|H}}|| T. Hida, "Brownian motion", Springer (1980) {{MR|0566333}} {{MR|0562914}} {{ZBL|0432.60002}} {{ZBL|0423.60063}}
 +
|-
 +
|valign="top"|{{Ref|S}}|| F. Spitzer, "Principles of random walk", v. Nostrand (1964) {{MR|0171290}} {{ZBL|0119.34304}}
 +
|-
 +
|valign="top"|{{Ref|Y}}|| J. Yeh, "Stochastic processes and the Wiener integral", M. Dekker (1973) {{MR|0474528}} {{ZBL|0277.60018}}
 +
|-
 +
|valign="top"|{{Ref|Do}}|| J.L. Doob, "Classical potential theory and its probabilistic counterpart", Springer (1984) {{MR|0731258}} {{ZBL|0549.31001}}
 +
|}

Latest revision as of 16:32, 14 January 2024


2020 Mathematics Subject Classification: Primary: 60J65 [MSN][ZBL]

A homogeneous Gaussian process $ X( t) $ with independent increments. A Wiener process serves as one of the models of Brownian motion. A simple transformation will convert a Wiener process into the "standard" Wiener process $ X( t) $, $ t \geq 0 $, for which

$$ X ( 0) = 0,\ \ {\mathsf E} ( X ( t) - X ( s)) = 0, $$

$$ {\mathsf D} [ X ( t) - X ( s)] = t - s,\ s \leq t. $$

For these average values and incremental variances, this is the only almost-surely continuous process with independent increments. In what follows, the Wiener process will be understood to be this process.

The Wiener process $ X( t) $, $ 0 \leq t \leq 1 $, can also be defined as the Gaussian process with zero expectation and covariance function

$$ B ( s, t) = \min ( s, t). $$

The Wiener process $ X= X( t) $, $ t \geq 0 $, may also be defined as the homogeneous Markov process with transition function

$$ P ( t, x, \Gamma ) = \int\limits _ \Gamma p ( t, x, y) dy, $$

where the transition density $ p( t, x, y) $ is the fundamental solution of the parabolic differential equation

$$ \frac{\partial p }{\partial t } = \ { \frac{1}{2} } \frac{\partial ^ {2} p }{\partial x ^ {2} } , $$

given by the formula

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

The transition function $ P( t, x, \Gamma ) $ is invariant with respect to translations in the phase space:

$$ P ( t, x + y, \Gamma ) = P ( t, x, \Gamma - y), $$

where $ \Gamma - y $ denotes the set $ \{ {z } : {z + y \in \Gamma } \} $.

The Wiener process is the continuous analogue of the random walk of a particle which, at discrete moments of time $ t= k \Delta t $( multiples of $ \Delta t $), is randomly displaced by a quantity $ \Delta X( t) $, independent of the past ( $ {\mathsf E} \Delta X( t) = 0 $, $ {\mathsf D} \Delta X( t) = \Delta t $); more precisely, if

$$ X ( t) = \sum _{k = 0 } ^ {m-1} \Delta X \left ( { \frac{k}{n} } \right ) + ( nt - m) \Delta X \left ( { \frac{m}{n} } \right ) ,\ \ 0 \leq t \leq 1, $$

is the random trajectory of the motion of such a particle on the interval $ [ 0, 1] $( where $ m= [ nt] $ is the integer part of $ nt $, $ X( t) = nt \Delta X ( 0) $ if $ 0 \leq t < 1/n $ and $ {\mathsf P} _ {n} $ is the corresponding probability distribution in the space of continuous functions $ x= x( t) $, $ 0 \leq t \leq 1 $), then the probability distribution $ {\mathsf P} $ of the trajectory of the Wiener process $ X( t) $, $ 0 \leq t \leq 1 $, is the limit (in the sense of weak convergence) of the distributions $ {\mathsf P} _ {n} $ as $ n \rightarrow \infty $.

As a function with values in the Hilbert space $ L _ {2} ( \Omega ) $ of all random variables $ X $ with $ {\mathsf E} X ^ {2} < \infty $, in which the scalar product is defined by the formula

$$ \langle X _ {1} , X _ {2} \rangle = {\mathsf E} X _ {1} X _ {2} , $$

the Wiener process $ X = X( t) $, $ 0 \leq t \leq 1 $, may be canonically represented as follows:

$$ X ( t) = \sum _ {k = 0 } ^ \infty z _ {k} \phi _ {k} ( t), $$

where $ z _ {k} $ are independent Gaussian variables:

$$ {\mathsf E} z _ {k} = 0,\ \ {\mathsf D} z _ {k} = \frac{1}{\left [ { \frac \pi {2} } ( 2k + 1) \right ] ^ {2} } , $$

and

$$ \phi _ {k} ( t) = \ \sin \left [ { \frac \pi {2} } ( 2k + 1) t \right ] ,\ \ k = 0, 1 \dots $$

are the eigenfunctions of the operator $ B $ defined by the formula

$$ B \phi ( t) = \int\limits _ { 0 } ^ { 1 } B ( s, t) \phi ( s) ds $$

in the Hilbert space $ L _ {2} [ 0, 1] $ of all square-integrable (with respect to Lebesgue measure) functions $ \phi $ on $ [ 0, 1] $.

Almost-all trajectories of the Wiener process have the following properties:

$$ {\lim\limits \sup } _ {h \rightarrow \infty } \ \frac{X ( h) }{\sqrt {2 h \mathop{\rm ln} \mathop{\rm ln} { \frac{1}{h} } } } = 1,\ \ X ( 0) = 0, $$

which is the law of the iterated logarithm;

$$ \lim\limits _ {h \rightarrow + 0 } \ \sup _ {0 \leq t \leq \delta - h } \ \frac{| X ( t+ h) - X ( t) | }{\sqrt {2 h \mathop{\rm ln} { \frac \delta {h} } } } = 1, $$

characterizing the modulus of continuity on $ [ 0, h] $; and

$$ \lim\limits _ {h \rightarrow \infty } \sum_{k = 0 } ^ {n-1} | \Delta X ( kh) | ^ {2} = \delta , $$

$$ h = \frac \delta {n} ,\ \Delta X ( t) = X ( t+ h) - X ( t). $$

When applied to the Wiener process $ X _ {1} ( t) = tX( 1/t) $, $ 0 \leq t < \infty $, the law of the iterated logarithm reads:

$$ {\lim\limits \sup } _ {t \rightarrow \infty } \frac{X ( t) }{\sqrt {2 t \mathop{\rm ln} \mathop{\rm ln} t } } = 1. $$

The distributions of the maximum $ \max _ {0 \leq s \leq t } X( s) $, of the time $ r _ {x} $ at which the Brownian particle first reaches a fixed point $ x > 0 $ and of the location $ \tau $ of the maximum $ \max _ {0 \leq s \leq t } X( s) $ give insight in the nature of the movement of a Brownian particle; these distributions are given by the following formulas:

$$ {\mathsf P} \left \{ \max _ {0 \leq s \leq t } X( s) \right \} = \frac{2}{\sqrt {2 \pi t } } \int\limits _ { x } ^ \infty e ^ {- u ^ {2} /2t } du , $$

$$ P \{ \tau _ {x} \geq t \} = P \left \{ \max _ {0 \leq s \leq t } X ( s) \leq x \right \} = $$

$$ = \ \sqrt { \frac{2} \pi } \int\limits _ { 0 } ^ { {x/ } \sqrt t } e ^ {- u ^ {2} /2 } d u , $$

while the simultaneous density of the maximum $ \max _ {0 \leq s \leq t } X( s) $ and its location $ \tau $ is given by:

$$ {\mathsf P} \{ \tau _ {x} \geq t \} = \ \sqrt { \frac{2} \pi } \int\limits _ { 0 } ^ { {x } / \sqrt t } e ^ {- u ^ {2} /2t } d u , $$

$$ 0 < s < t ,\ 0 \leq x < \infty . $$

(The laws of the Wiener process remain unchanged on transforming the phase space $ x \rightarrow - x $.) The joint distribution of the maximum point $ \tau $, $ 0 \leq \tau \leq t $, and of the maximum $ \max _ {0\leq s \leq t } X( s) $ itself has the probability density

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

$$ 0 \leq s \leq t,\ 0 \leq x < \infty , $$

while the point $ \tau $ by itself (with probability one there is only one maximum on the interval $ 0 \leq s \leq t $) is distributed according to the arcsine law:

$$ {\mathsf P} \{ \tau \leq s \} = \ { \frac{2} \pi } \mathop{\rm arc} \sin \sqrt { \frac{s}{t} } ,\ \ 0 \leq s \leq t, $$

with the probability density:

$$ p ( s) = { \frac{1}{\pi \sqrt s( t- s) } } ,\ \ 0 \leq s \leq t. $$

The following properties of the Wiener process are readily deduced from the formulas given above. The Brownian trajectory is nowhere differentiable; on starting from any point $ x $ this trajectory crosses the "level" $ x $( returns to its initial point) infinitely many times with probability one, however short the time $ \delta $; the Brownian trajectory passes through all points $ x $( more precisely, $ \tau _ {x} < \infty $) with probability one (the most probable value of $ \tau _ {x} $ is of the order $ x ^ {2} $ for large $ x $); this trajectory, if considered on a fixed interval $ [ 0, t] $, tends to attain the extremal values near the end-points $ s= 0 $ and $ s= t $.

Since a Wiener process is a homogeneous Markov process, there exists an invariant measure $ Q( d x) $ for it, namely:

$$ Q ( A) \equiv \int\limits Q ( dx) P ( t, x, A), $$

which, since the transition function $ P( t, x, A) $ has been seen to be invariant, coincides with the Lebesgue measure on the real line: $ Q( d x) = d x $. The time $ T( A) $ which a Brownian particle spends in $ A $ between the times 0 and $ T $ is such that

$$ \frac{T ( A _ {1} ) }{T ( A _ {2} ) } \rightarrow \ \frac{Q ( A _ {1} ) }{Q ( A _ {2} ) } $$

as $ T \rightarrow \infty $, with probability one for any bounded Borel sets $ A _ {1} $ and $ A _ {2} $.

Wiener random fields, introduced by P. Lévy [L], are analogues of the Wiener process $ X = X( t) $ for a vector parameter $ t = ( t _ {1} \dots t _ {n} ) $.

References

[IM] K. Itô, H.P. McKean jr., "Diffusion processes and their sample paths" , Springer (1974) MR0345224 Zbl 0285.60063
[PR] Yu.V. Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian) MR0251754
[L] P. Lévy, "Processus stochastiques et mouvement Brownien" , Gauthier-Villars (1965) MR0190953 Zbl 0137.11602
[P] V.P. Pavlov, "Brownian motion" , Large Soviet Encyclopaedia , 4 (In Russian)

Comments

The Wiener process is more commonly referred to as Brownian motion in the Western literature. It is by far the most important construct in stochastic analysis. See [Du][RY] for up-to-date accounts of its properties. Of particular importance is the theory of local time. The occupation time of a Borel set $ B \subset \mathbf R $ on the interval $ [ 0, t] $ is:

$$ \Gamma _ {t} ( B) = \int\limits _ { 0 } ^ { t } I _ {B} ( X( s)) ds . $$

There exists an almost-surely jointly-continuous random field $ L ( t, x) $ for $ ( t, x) \in \mathbf R _ {+} \times \mathbf R $ such that

$$ \Gamma _ {t} ( B) = 2 \int\limits _ { B } L( t, x) dx; $$

$ L( t, x) $ is the local time at $ x $. For fixed $ x \in \mathbf R $, sample paths of the process $ t \mapsto L( t, x) $ are increasing and continuous but singular with respect to Lebesgue measure.

See also Markov process; Stochastic differential equation.

References

[Du] R. Durrett, "Brownian motion and martingales in analysis", Wadsworth (1984) MR0750829 Zbl 0554.60075
[KS] I. Karatzas, S.E. Shreve, "Brownian motion and stochastic calculus", Springer (1988) MR0917065 Zbl 0638.60065
[RY] D. Revuz, M. Yor, "Continuous martingales and Brownian motion", Springer (1990) MR1725357 MR1303781 MR1083357 Zbl 1087.60040 Zbl 0917.60006 Zbl 0804.60001 Zbl 0731.60002
[Dy] E.B. Dynkin, "Markov processes", 1, Springer (1965) (Translated from Russian) MR0193671 Zbl 0132.37901
[F] W. Feller, "An introduction to probability theory and its applications", 1–2, Wiley (1968–1971)
[GS] I.I. Gihman, A.V. Skorohod, "The theory of stochastic processes", III, Springer (1975) (Translated from Russian) MR0375463 Zbl 0305.60027
[H] T. Hida, "Brownian motion", Springer (1980) MR0566333 MR0562914 Zbl 0432.60002 Zbl 0423.60063
[S] F. Spitzer, "Principles of random walk", v. Nostrand (1964) MR0171290 Zbl 0119.34304
[Y] J. Yeh, "Stochastic processes and the Wiener integral", M. Dekker (1973) MR0474528 Zbl 0277.60018
[Do] J.L. Doob, "Classical potential theory and its probabilistic counterpart", Springer (1984) MR0731258 Zbl 0549.31001
How to Cite This Entry:
Wiener process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wiener_process&oldid=24712
This article was adapted from an original article by Yu.A. Rozanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article