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A generalization of the notion of [[Wiener process|Wiener process]] for the case of multivariate time. This generalization can be performed in two ways.
 
A generalization of the notion of [[Wiener process|Wiener process]] for the case of multivariate time. This generalization can be performed in two ways.
  
==<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w1201801.png" />-parameter Wiener field (Brownian motion).==
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==$N$-parameter Wiener field (Brownian motion).==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w1201802.png" /> be a Gaussian separable real-valued field on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w1201803.png" /> with zero mean and covariance function
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Let $W ^ { ( N ) } ( t )$ be a Gaussian separable real-valued field on $\mathbf R _ { + } ^ { N } = \{ t = ( t _ { 1 } , \dots , t _ { N } ) : t _ { i } \geq 0 \}$ with zero mean and covariance 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/w120/w120180/w1201804.png" /></td> </tr></table>
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\begin{equation*} \mathsf{E} W ^ { ( N ) } ( t ) W ^ { ( N ) } ( s ) = \prod _ { i = 1 } ^ { N } t _ { i } \bigwedge s _ { i }, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w1201805.png" />. Such a field can be regarded as the distribution function of a [[White noise|white noise]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w1201806.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w1201807.png" />, which is a [[Random function|random function]] on bounded Borel sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w1201808.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w1201809.png" /> has a [[Normal distribution|normal distribution]] with zero mean and covariance function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018010.png" /> [[#References|[a11]]]. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018011.png" /> denotes the [[Lebesgue measure|Lebesgue measure]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018012.png" />. The following equality holds: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018013.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018014.png" /> is a parallelepiped in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018015.png" />.
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where $t \wedge s = \operatorname { min } ( t , s )$. Such a field can be regarded as the distribution function of a [[White noise|white noise]] $W (\cdot)$ on $\mathbf{R} ^ { N }$, which is a [[Random function|random function]] on bounded Borel sets in $\mathbf{R} ^ { N }$ such that $W ( A )$ has a [[Normal distribution|normal distribution]] with zero mean and covariance function $\mathsf{E} W ( A ) W ( B ) = m ( A \cap B )$ [[#References|[a11]]]. Here, $m ( . )$ denotes the [[Lebesgue measure|Lebesgue measure]] on $\mathbf{R} ^ { N }$. The following equality holds: $W ^ { ( N ) } ( t ) = W ( R _ { t } )$, where $R _ { t } = \prod _ { i = 1 } ^ { N } [ 0 , t _ { i } )$ is a parallelepiped in ${\bf R} _ { + } ^ { N }$.
  
The random field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018016.png" /> was introduced by T. Kitagava [[#References|[a14]]] in connection with its applications to statistical problems. N.N. Chentsov proved the almost sure continuity of the sample functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018017.png" /> [[#References|[a3]]]. For any fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018018.png" /> time variables, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018019.png" /> is a one-parameter Wiener process as a function of the free time variable. Some properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018020.png" /> are similar to the corresponding properties of the Wiener process: the sample functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018021.png" /> almost surely satisfy Hölder's stochastic condition with exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018022.png" /> [[#References|[a2]]]; various forms of the [[Law of the iterated logarithm|law of the iterated logarithm]] hold true ([[#References|[a5]]], [[#References|[a18]]], [[#References|[a20]]]). An exact formula for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018023.png" /> exists only for the Wiener process. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018024.png" />, only lower and upper bounds (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018025.png" />, [[#References|[a12]]]) and some asymptotic formulas for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018026.png" /> [[#References|[a21]]] have been derived so far (1998). The level sets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018027.png" /> have an extremely complicated geometric and topological structure ([[#References|[a8]]], [[#References|[a9]]], [[#References|[a10]]], [[#References|[a13]]]). R.J. Adler [[#References|[a1]]] showed that the [[Hausdorff dimension|Hausdorff dimension]] of these sets equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018028.png" />.
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The random field $W ^ { ( N ) } ( t )$ was introduced by T. Kitagava [[#References|[a14]]] in connection with its applications to statistical problems. N.N. Chentsov proved the almost sure continuity of the sample functions of $W ^ { ( N ) } ( t )$ [[#References|[a3]]]. For any fixed $N - 1$ time variables, $W ^ { ( N ) } ( t )$ is a one-parameter Wiener process as a function of the free time variable. Some properties of $W ^ { ( N ) } ( t )$ are similar to the corresponding properties of the Wiener process: the sample functions of $W ^ { ( N ) } ( t )$ almost surely satisfy Hölder's stochastic condition with exponent $\alpha &lt; 1 / 2$ [[#References|[a2]]]; various forms of the [[Law of the iterated logarithm|law of the iterated logarithm]] hold true ([[#References|[a5]]], [[#References|[a18]]], [[#References|[a20]]]). An exact formula for $\mathsf{P} \{ \operatorname { sup } W ^ { ( N ) } ( t ) &gt; u \}$ exists only for the Wiener process. For $N &gt; 1$, only lower and upper bounds ($N = 2$, [[#References|[a12]]]) and some asymptotic formulas for $u \rightarrow \infty$ [[#References|[a21]]] have been derived so far (1998). The level sets of $W ^ { ( N ) } ( t )$ have an extremely complicated geometric and topological structure ([[#References|[a8]]], [[#References|[a9]]], [[#References|[a10]]], [[#References|[a13]]]). R.J. Adler [[#References|[a1]]] showed that the [[Hausdorff dimension|Hausdorff dimension]] of these sets equals $N - 1 / 2$.
  
The Wiener process is a [[Markov process|Markov process]]: conditional of the present value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018029.png" />, the past <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018030.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018031.png" />) and the future <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018032.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018033.png" />) are independent. For the multivariate case there are several definitions of the Markovian property. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018034.png" /> denote a family of Jordan surfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018035.png" />. Each such surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018036.png" /> divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018037.png" /> into two parts: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018038.png" />, the interior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018039.png" />, or the "past" , and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018040.png" />, the exterior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018041.png" />, or the "future" . A random field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018042.png" /> is said to be Markovian with respect to the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018043.png" /> if for arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018044.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018045.png" /> and arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018047.png" />, the random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018049.png" /> are conditionally independent given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018050.png" /> [[#References|[a24]]]. A Wiener field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018051.png" /> is a Markovian field with respect to the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018052.png" /> consisting of all finite unions of rectangles whose sides are parallel to the coordinate axes ([[#References|[a22]]], [[#References|[a23]]]). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018053.png" />, its sharp field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018054.png" /> and germ field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018055.png" /> are defined, respectively, by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018057.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018058.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018059.png" />-neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018060.png" />. A Wiener sheet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018061.png" /> is germ Markovian, i.e. for every bounded subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018062.png" />, the fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018063.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018064.png" /> are conditionally independent given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018065.png" /> ([[#References|[a6]]], [[#References|[a7]]], [[#References|[a22]]]).
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The Wiener process is a [[Markov process|Markov process]]: conditional of the present value $W ( t )$, the past $W ( v )$ ($v &lt; t$) and the future $W ( u )$ ($u &gt; t$) are independent. For the multivariate case there are several definitions of the Markovian property. Let $\mathcal{M}$ denote a family of Jordan surfaces in $\mathbf{R} ^ { N }$. Each such surface $\partial D$ divides $\mathbf{R} ^ { N }$ into two parts: $D ^ { - }$, the interior of $\partial D$, or the "past" , and $D ^ { + }$, the exterior of $\partial D$, or the "future" . A random field $X ( t )$ is said to be Markovian with respect to the family $\mathcal{M}$ if for arbitrary $\partial D$ from $\mathcal{M}$ and arbitrary $t _ { 1 } \in D ^ { - }$, $t _ { 2 } \in D ^ { + }$, the random variables $X ( t _ { 1 } )$ and $X ( t _ { 2 } )$ are conditionally independent given $\{ X ( t ) : t \in \partial D \}$ [[#References|[a24]]]. A Wiener field $W ^ { ( 2 ) } ( t )$ is a Markovian field with respect to the family $\mathcal{M}$ consisting of all finite unions of rectangles whose sides are parallel to the coordinate axes ([[#References|[a22]]], [[#References|[a23]]]). For $A \subset \mathbf{R} ^ { 2 }$, its sharp field $H ( A )$ and germ field $G ( A )$ are defined, respectively, by $H ( A ) = \sigma \left\{ W ^ { ( 2 ) } ( t ) : t \in A \right\}$ and $G ( A ) = \cap _ { \epsilon  &gt; 0} H ( A _ { \epsilon } )$, where $A _ { \epsilon }$ is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018059.png"/>-neighbourhood of $A$. A Wiener sheet $W ^ { ( 2 ) } ( t )$ is germ Markovian, i.e. for every bounded subset $A \subset \mathbf{R} _ { + } ^ { 2 }$, the fields $H ( A )$ and $H ( A ^ { c } )$ are conditionally independent given $G ( \partial A )$ ([[#References|[a6]]], [[#References|[a7]]], [[#References|[a22]]]).
  
Among the objects closely related to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018066.png" /> are the Wiener pillow and the Wiener bridge. These are Gaussian random fields (cf. also [[Random field|Random field]]) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018067.png" /> with zero mean and covariance functions
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Among the objects closely related to $W ^ { ( N ) } ( t )$ are the Wiener pillow and the Wiener bridge. These are Gaussian random fields (cf. also [[Random field|Random field]]) on $[ 0,1 ] ^ { N }$ with zero mean and covariance functions
  
<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/w120/w120180/w12018068.png" /></td> </tr></table>
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\begin{equation*} r _ { 1 } ( t , s ) = \prod _ { i = 1 } ^ { N } ( t _ { i } \bigwedge s _ { i } - t _ { i } s _ { i } ), \end{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/w120/w120180/w12018069.png" /></td> </tr></table>
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\begin{equation*} r _ { 2 } ( t , s ) = \prod _ { i = 1 } ^ { N } t _ { i } \wedge s _ { i } - \prod _ { i = 1 } ^ { N } t _ { i } s _ { i } , \end{equation*}
  
 
respectively.
 
respectively.
  
==Lévy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018070.png" />-parameter Brownian motion.==
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==Lévy $N$-parameter Brownian motion.==
This is a Gaussian random field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018071.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018072.png" /> with zero mean and covariance function
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This is a Gaussian random field $\xi ( t )$ on $\mathbf{R} ^ { N }$ with zero mean and covariance 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/w120/w120180/w12018073.png" /></td> </tr></table>
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\begin{equation*} \mathsf{E} \xi ( t ) \xi ( s ) = \frac { 1 } { 2 } ( | t | + | s | - | t - s | ), \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018074.png" /> [[#References|[a15]]]. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018076.png" /> becomes a [[Wiener process|Wiener process]]. The random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018077.png" /> clearly form a Wiener process if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018078.png" /> moves along some semi-straight line with terminal point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018079.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018080.png" /> has the following representation in terms of white noise:
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where $| t | = \sqrt { \sum _ { k = 1 } ^ { N } t _ { k } ^ { 2 } }$ [[#References|[a15]]]. When $N = 1$, $\xi ( t )$ becomes a [[Wiener process|Wiener process]]. The random variables $\xi ( t ) - \xi ( s )$ clearly form a Wiener process if $t$ moves along some semi-straight line with terminal point $s$. $\xi ( t )$ has the following representation in terms of white noise:
  
<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/w120/w120180/w12018081.png" /></td> </tr></table>
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\begin{equation*} \xi ( t ) = \frac { 1 } { \sqrt { \omega _ { N + 1 } } } \int _ { \mathbf{R} ^ { N } } \frac { e ^ { i ( t , \lambda ) } - 1 } { | \lambda | ^ { ( N + 1 ) / 2 } } W ( d \lambda ), \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018082.png" /> is the surface area of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018083.png" />-dimensional unit sphere [[#References|[a17]]]. H.P. McKean Jr. [[#References|[a16]]] has shown that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018084.png" /> is germ Markovian with respect to closed bounded subsets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018085.png" /> for each odd <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018086.png" />, whereas for each even <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120180/w12018087.png" /> the Markovian property does not hold.
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where $\omega _ { n }$ is the surface area of the $n$-dimensional unit sphere [[#References|[a17]]]. H.P. McKean Jr. [[#References|[a16]]] has shown that $\xi ( t )$ is germ Markovian with respect to closed bounded subsets in $\mathbf{R} ^ { N } \backslash \{ 0 \}$ for each odd $N$, whereas for each even $N$ the Markovian property does not hold.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R.J. Adler, "The uniform dimension of the level sets of a Brownian sheet" ''Ann. of Probab.'' , '''6''' (1978) pp. 509–518 {{MR|0490818}} {{ZBL|0378.60028}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R.J. Adler, "The geometry of random fields" , Wiley (1981) {{MR|0611857}} {{ZBL|0478.60059}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> N.N. Chentsov, "Wiener random fields depending on several parameters" ''Dokl. Akad. Nauk SSSR'' , '''106''' (1956) pp. 607–609</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> N.N. Chentsov, "A multiparametric Brownian motion Lévy and generalized white noise" ''Theory Probab. Appl.'' , '''2''' (1957) pp. 281–282</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> M. Csőrgő, P. Révész, "Strong approximations in probability and statistics" , Akad. Kiado (1981) {{MR|0666546}} {{ZBL|0539.60029}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> R.C. Dalang, F. Russo, "A prediction problem for the Brownian sheet" ''J. Multivariate Anal.'' , '''26''' (1988) pp. 16–47 {{MR|0955202}} {{ZBL|0664.60052}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> R.C. Dalang, J.B. Walsh, "The sharp Markov property of the Brownian sheet and related processes" ''Acta Math.'' , '''168''' (1992) pp. 153–218 {{MR|1161265}} {{ZBL|0759.60056}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> R.C. Dalang, J.B. Walsh, "Geography of the level sets of the Brownian sheet" ''Probab. Th. Rel. Fields'' , '''96''' (1993) pp. 153–176 {{MR|1227030}} {{ZBL|0792.60038}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> R.C. Dalang, J.B. Walsh, "The structure of a Brownian bubble" ''Probab. Th. Rel. Fields'' , '''96''' (1993) pp. 475–501 {{MR|1234620}} {{ZBL|0794.60047}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> R.C. Dalang, T. Mountford, "Nondifferentiability of curves on the Brownian sheet" ''Ann. of Probab.'' , '''24''' (1996) pp. 182–195 {{MR|1387631}} {{ZBL|0861.60058}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> R.M. Dudley, "Sample functions of the Gaussian process" ''Ann. of Probab.'' , '''1''' (1973) pp. 66–103 {{MR|0346884}} {{ZBL|0261.60033}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> V. Goodman, "Distribution estimates for functionals of the two-parameter Wiener process" ''Ann. of Probab.'' , '''4''' (1976) pp. 977–982 {{MR|0423556}} {{ZBL|0344.60048}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> W. Kendall, "Contours of Brownian processes with several-dimensional time" ''ZWvG'' , '''52''' (1980) pp. 269–276 {{MR|0576887}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> T. Kitagava, "Analysis of variance applied to function spaces" ''Mem. Fac. Sci. Kyushu Univ. Ser. A'' , '''6''' (1951) pp. 41–53</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> P. Lévy, "Processes stochastiques et mouvement brownien" , Gauthier-Villars (1948) {{MR|}} {{ZBL|0137.11602}} </TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top"> H.P. McKean Jr., "Brownian motion with a several-dimensional time" ''Theory Probab. Appl.'' , '''8''' (1963) pp. 335–354</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top"> G.M. Molchan, "Some problems for Lévy's Brownian motion" ''Theory Probab. Appl.'' , '''12''' (1967) pp. 682–690</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top"> S. Orey, W. Pruitt, "Sample functions of the N-parameter Wiener process" ''Ann. of Probab.'' , '''1''' (1973) pp. 138–163 {{MR|0346925}} {{ZBL|0284.60036}} </TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top"> S.R. Paranjape, C. Park, "Distribution of the supremum of the two-parameter Yeh–Wiener process on the boundary" ''J. Appl. Probab.'' , '''10''' (1973) pp. 875–880 {{MR|0381015}} {{ZBL|0281.60081}} </TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top"> S.R. Paranjape, C. Park, "Laws of iterated logarithm of multiparameter Wiener process" ''J. Multivariate Anal.'' , '''3''' (1973) pp. 132–136 {{MR|0326852}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top"> V.I. Piterbarg, "Asymptotic methods in the theory of Gaussian processes and fields" , Amer. Math. Soc. (1996) {{MR|1361884}} {{ZBL|0841.60024}} </TD></TR><TR><TD valign="top">[a22]</TD> <TD valign="top"> Yu.A. Rosanov, "Markov random fields" , Springer (1982)</TD></TR><TR><TD valign="top">[a23]</TD> <TD valign="top"> J.B. Walsh, "Propagation of singularities in the Brownian sheet" ''Ann. of Probab.'' , '''Ann. 10''' (1982) pp. 279–288 {{MR|0647504}} {{ZBL|0528.60076}} </TD></TR><TR><TD valign="top">[a24]</TD> <TD valign="top"> M.I. Yadrenko, "Spectral theory of random fields" , Optim. Software (1983) {{MR|0697386}} {{ZBL|0539.60048}} </TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top"> R.J. Adler, "The uniform dimension of the level sets of a Brownian sheet" ''Ann. of Probab.'' , '''6''' (1978) pp. 509–518 {{MR|0490818}} {{ZBL|0378.60028}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top"> R.J. Adler, "The geometry of random fields" , Wiley (1981) {{MR|0611857}} {{ZBL|0478.60059}} </td></tr><tr><td valign="top">[a3]</td> <td valign="top"> N.N. Chentsov, "Wiener random fields depending on several parameters" ''Dokl. Akad. Nauk SSSR'' , '''106''' (1956) pp. 607–609</td></tr><tr><td valign="top">[a4]</td> <td valign="top"> N.N. Chentsov, "A multiparametric Brownian motion Lévy and generalized white noise" ''Theory Probab. Appl.'' , '''2''' (1957) pp. 281–282</td></tr><tr><td valign="top">[a5]</td> <td valign="top"> M. Csőrgő, P. Révész, "Strong approximations in probability and statistics" , Akad. Kiado (1981) {{MR|0666546}} {{ZBL|0539.60029}} </td></tr><tr><td valign="top">[a6]</td> <td valign="top"> R.C. Dalang, F. Russo, "A prediction problem for the Brownian sheet" ''J. Multivariate Anal.'' , '''26''' (1988) pp. 16–47 {{MR|0955202}} {{ZBL|0664.60052}} </td></tr><tr><td valign="top">[a7]</td> <td valign="top"> R.C. Dalang, J.B. Walsh, "The sharp Markov property of the Brownian sheet and related processes" ''Acta Math.'' , '''168''' (1992) pp. 153–218 {{MR|1161265}} {{ZBL|0759.60056}} </td></tr><tr><td valign="top">[a8]</td> <td valign="top"> R.C. Dalang, J.B. Walsh, "Geography of the level sets of the Brownian sheet" ''Probab. Th. Rel. Fields'' , '''96''' (1993) pp. 153–176 {{MR|1227030}} {{ZBL|0792.60038}} </td></tr><tr><td valign="top">[a9]</td> <td valign="top"> R.C. Dalang, J.B. Walsh, "The structure of a Brownian bubble" ''Probab. Th. Rel. Fields'' , '''96''' (1993) pp. 475–501 {{MR|1234620}} {{ZBL|0794.60047}} </td></tr><tr><td valign="top">[a10]</td> <td valign="top"> R.C. Dalang, T. Mountford, "Nondifferentiability of curves on the Brownian sheet" ''Ann. of Probab.'' , '''24''' (1996) pp. 182–195 {{MR|1387631}} {{ZBL|0861.60058}} </td></tr><tr><td valign="top">[a11]</td> <td valign="top"> R.M. Dudley, "Sample functions of the Gaussian process" ''Ann. of Probab.'' , '''1''' (1973) pp. 66–103 {{MR|0346884}} {{ZBL|0261.60033}} </td></tr><tr><td valign="top">[a12]</td> <td valign="top"> V. Goodman, "Distribution estimates for functionals of the two-parameter Wiener process" ''Ann. of Probab.'' , '''4''' (1976) pp. 977–982 {{MR|0423556}} {{ZBL|0344.60048}} </td></tr><tr><td valign="top">[a13]</td> <td valign="top"> W. Kendall, "Contours of Brownian processes with several-dimensional time" ''ZWvG'' , '''52''' (1980) pp. 269–276 {{MR|0576887}} {{ZBL|}} </td></tr><tr><td valign="top">[a14]</td> <td valign="top"> T. Kitagava, "Analysis of variance applied to function spaces" ''Mem. Fac. Sci. Kyushu Univ. Ser. A'' , '''6''' (1951) pp. 41–53</td></tr><tr><td valign="top">[a15]</td> <td valign="top"> P. Lévy, "Processes stochastiques et mouvement brownien" , Gauthier-Villars (1948) {{MR|}} {{ZBL|0137.11602}} </td></tr><tr><td valign="top">[a16]</td> <td valign="top"> H.P. McKean Jr., "Brownian motion with a several-dimensional time" ''Theory Probab. Appl.'' , '''8''' (1963) pp. 335–354</td></tr><tr><td valign="top">[a17]</td> <td valign="top"> G.M. Molchan, "Some problems for Lévy's Brownian motion" ''Theory Probab. Appl.'' , '''12''' (1967) pp. 682–690</td></tr><tr><td valign="top">[a18]</td> <td valign="top"> S. Orey, W. Pruitt, "Sample functions of the N-parameter Wiener process" ''Ann. of Probab.'' , '''1''' (1973) pp. 138–163 {{MR|0346925}} {{ZBL|0284.60036}} </td></tr><tr><td valign="top">[a19]</td> <td valign="top"> S.R. Paranjape, C. Park, "Distribution of the supremum of the two-parameter Yeh–Wiener process on the boundary" ''J. Appl. Probab.'' , '''10''' (1973) pp. 875–880 {{MR|0381015}} {{ZBL|0281.60081}} </td></tr><tr><td valign="top">[a20]</td> <td valign="top"> S.R. Paranjape, C. Park, "Laws of iterated logarithm of multiparameter Wiener process" ''J. Multivariate Anal.'' , '''3''' (1973) pp. 132–136 {{MR|0326852}} {{ZBL|}} </td></tr><tr><td valign="top">[a21]</td> <td valign="top"> V.I. Piterbarg, "Asymptotic methods in the theory of Gaussian processes and fields" , Amer. Math. Soc. (1996) {{MR|1361884}} {{ZBL|0841.60024}} </td></tr><tr><td valign="top">[a22]</td> <td valign="top"> Yu.A. Rosanov, "Markov random fields" , Springer (1982)</td></tr><tr><td valign="top">[a23]</td> <td valign="top"> J.B. Walsh, "Propagation of singularities in the Brownian sheet" ''Ann. of Probab.'' , '''Ann. 10''' (1982) pp. 279–288 {{MR|0647504}} {{ZBL|0528.60076}} </td></tr><tr><td valign="top">[a24]</td> <td valign="top"> M.I. Yadrenko, "Spectral theory of random fields" , Optim. Software (1983) {{MR|0697386}} {{ZBL|0539.60048}} </td></tr></table>

Revision as of 17:00, 1 July 2020

A generalization of the notion of Wiener process for the case of multivariate time. This generalization can be performed in two ways.

$N$-parameter Wiener field (Brownian motion).

Let $W ^ { ( N ) } ( t )$ be a Gaussian separable real-valued field on $\mathbf R _ { + } ^ { N } = \{ t = ( t _ { 1 } , \dots , t _ { N } ) : t _ { i } \geq 0 \}$ with zero mean and covariance function

\begin{equation*} \mathsf{E} W ^ { ( N ) } ( t ) W ^ { ( N ) } ( s ) = \prod _ { i = 1 } ^ { N } t _ { i } \bigwedge s _ { i }, \end{equation*}

where $t \wedge s = \operatorname { min } ( t , s )$. Such a field can be regarded as the distribution function of a white noise $W (\cdot)$ on $\mathbf{R} ^ { N }$, which is a random function on bounded Borel sets in $\mathbf{R} ^ { N }$ such that $W ( A )$ has a normal distribution with zero mean and covariance function $\mathsf{E} W ( A ) W ( B ) = m ( A \cap B )$ [a11]. Here, $m ( . )$ denotes the Lebesgue measure on $\mathbf{R} ^ { N }$. The following equality holds: $W ^ { ( N ) } ( t ) = W ( R _ { t } )$, where $R _ { t } = \prod _ { i = 1 } ^ { N } [ 0 , t _ { i } )$ is a parallelepiped in ${\bf R} _ { + } ^ { N }$.

The random field $W ^ { ( N ) } ( t )$ was introduced by T. Kitagava [a14] in connection with its applications to statistical problems. N.N. Chentsov proved the almost sure continuity of the sample functions of $W ^ { ( N ) } ( t )$ [a3]. For any fixed $N - 1$ time variables, $W ^ { ( N ) } ( t )$ is a one-parameter Wiener process as a function of the free time variable. Some properties of $W ^ { ( N ) } ( t )$ are similar to the corresponding properties of the Wiener process: the sample functions of $W ^ { ( N ) } ( t )$ almost surely satisfy Hölder's stochastic condition with exponent $\alpha < 1 / 2$ [a2]; various forms of the law of the iterated logarithm hold true ([a5], [a18], [a20]). An exact formula for $\mathsf{P} \{ \operatorname { sup } W ^ { ( N ) } ( t ) > u \}$ exists only for the Wiener process. For $N > 1$, only lower and upper bounds ($N = 2$, [a12]) and some asymptotic formulas for $u \rightarrow \infty$ [a21] have been derived so far (1998). The level sets of $W ^ { ( N ) } ( t )$ have an extremely complicated geometric and topological structure ([a8], [a9], [a10], [a13]). R.J. Adler [a1] showed that the Hausdorff dimension of these sets equals $N - 1 / 2$.

The Wiener process is a Markov process: conditional of the present value $W ( t )$, the past $W ( v )$ ($v < t$) and the future $W ( u )$ ($u > t$) are independent. For the multivariate case there are several definitions of the Markovian property. Let $\mathcal{M}$ denote a family of Jordan surfaces in $\mathbf{R} ^ { N }$. Each such surface $\partial D$ divides $\mathbf{R} ^ { N }$ into two parts: $D ^ { - }$, the interior of $\partial D$, or the "past" , and $D ^ { + }$, the exterior of $\partial D$, or the "future" . A random field $X ( t )$ is said to be Markovian with respect to the family $\mathcal{M}$ if for arbitrary $\partial D$ from $\mathcal{M}$ and arbitrary $t _ { 1 } \in D ^ { - }$, $t _ { 2 } \in D ^ { + }$, the random variables $X ( t _ { 1 } )$ and $X ( t _ { 2 } )$ are conditionally independent given $\{ X ( t ) : t \in \partial D \}$ [a24]. A Wiener field $W ^ { ( 2 ) } ( t )$ is a Markovian field with respect to the family $\mathcal{M}$ consisting of all finite unions of rectangles whose sides are parallel to the coordinate axes ([a22], [a23]). For $A \subset \mathbf{R} ^ { 2 }$, its sharp field $H ( A )$ and germ field $G ( A )$ are defined, respectively, by $H ( A ) = \sigma \left\{ W ^ { ( 2 ) } ( t ) : t \in A \right\}$ and $G ( A ) = \cap _ { \epsilon > 0} H ( A _ { \epsilon } )$, where $A _ { \epsilon }$ is an -neighbourhood of $A$. A Wiener sheet $W ^ { ( 2 ) } ( t )$ is germ Markovian, i.e. for every bounded subset $A \subset \mathbf{R} _ { + } ^ { 2 }$, the fields $H ( A )$ and $H ( A ^ { c } )$ are conditionally independent given $G ( \partial A )$ ([a6], [a7], [a22]).

Among the objects closely related to $W ^ { ( N ) } ( t )$ are the Wiener pillow and the Wiener bridge. These are Gaussian random fields (cf. also Random field) on $[ 0,1 ] ^ { N }$ with zero mean and covariance functions

\begin{equation*} r _ { 1 } ( t , s ) = \prod _ { i = 1 } ^ { N } ( t _ { i } \bigwedge s _ { i } - t _ { i } s _ { i } ), \end{equation*}

\begin{equation*} r _ { 2 } ( t , s ) = \prod _ { i = 1 } ^ { N } t _ { i } \wedge s _ { i } - \prod _ { i = 1 } ^ { N } t _ { i } s _ { i } , \end{equation*}

respectively.

Lévy $N$-parameter Brownian motion.

This is a Gaussian random field $\xi ( t )$ on $\mathbf{R} ^ { N }$ with zero mean and covariance function

\begin{equation*} \mathsf{E} \xi ( t ) \xi ( s ) = \frac { 1 } { 2 } ( | t | + | s | - | t - s | ), \end{equation*}

where $| t | = \sqrt { \sum _ { k = 1 } ^ { N } t _ { k } ^ { 2 } }$ [a15]. When $N = 1$, $\xi ( t )$ becomes a Wiener process. The random variables $\xi ( t ) - \xi ( s )$ clearly form a Wiener process if $t$ moves along some semi-straight line with terminal point $s$. $\xi ( t )$ has the following representation in terms of white noise:

\begin{equation*} \xi ( t ) = \frac { 1 } { \sqrt { \omega _ { N + 1 } } } \int _ { \mathbf{R} ^ { N } } \frac { e ^ { i ( t , \lambda ) } - 1 } { | \lambda | ^ { ( N + 1 ) / 2 } } W ( d \lambda ), \end{equation*}

where $\omega _ { n }$ is the surface area of the $n$-dimensional unit sphere [a17]. H.P. McKean Jr. [a16] has shown that $\xi ( t )$ is germ Markovian with respect to closed bounded subsets in $\mathbf{R} ^ { N } \backslash \{ 0 \}$ for each odd $N$, whereas for each even $N$ the Markovian property does not hold.

References

[a1] R.J. Adler, "The uniform dimension of the level sets of a Brownian sheet" Ann. of Probab. , 6 (1978) pp. 509–518 MR0490818 Zbl 0378.60028
[a2] R.J. Adler, "The geometry of random fields" , Wiley (1981) MR0611857 Zbl 0478.60059
[a3] N.N. Chentsov, "Wiener random fields depending on several parameters" Dokl. Akad. Nauk SSSR , 106 (1956) pp. 607–609
[a4] N.N. Chentsov, "A multiparametric Brownian motion Lévy and generalized white noise" Theory Probab. Appl. , 2 (1957) pp. 281–282
[a5] M. Csőrgő, P. Révész, "Strong approximations in probability and statistics" , Akad. Kiado (1981) MR0666546 Zbl 0539.60029
[a6] R.C. Dalang, F. Russo, "A prediction problem for the Brownian sheet" J. Multivariate Anal. , 26 (1988) pp. 16–47 MR0955202 Zbl 0664.60052
[a7] R.C. Dalang, J.B. Walsh, "The sharp Markov property of the Brownian sheet and related processes" Acta Math. , 168 (1992) pp. 153–218 MR1161265 Zbl 0759.60056
[a8] R.C. Dalang, J.B. Walsh, "Geography of the level sets of the Brownian sheet" Probab. Th. Rel. Fields , 96 (1993) pp. 153–176 MR1227030 Zbl 0792.60038
[a9] R.C. Dalang, J.B. Walsh, "The structure of a Brownian bubble" Probab. Th. Rel. Fields , 96 (1993) pp. 475–501 MR1234620 Zbl 0794.60047
[a10] R.C. Dalang, T. Mountford, "Nondifferentiability of curves on the Brownian sheet" Ann. of Probab. , 24 (1996) pp. 182–195 MR1387631 Zbl 0861.60058
[a11] R.M. Dudley, "Sample functions of the Gaussian process" Ann. of Probab. , 1 (1973) pp. 66–103 MR0346884 Zbl 0261.60033
[a12] V. Goodman, "Distribution estimates for functionals of the two-parameter Wiener process" Ann. of Probab. , 4 (1976) pp. 977–982 MR0423556 Zbl 0344.60048
[a13] W. Kendall, "Contours of Brownian processes with several-dimensional time" ZWvG , 52 (1980) pp. 269–276 MR0576887
[a14] T. Kitagava, "Analysis of variance applied to function spaces" Mem. Fac. Sci. Kyushu Univ. Ser. A , 6 (1951) pp. 41–53
[a15] P. Lévy, "Processes stochastiques et mouvement brownien" , Gauthier-Villars (1948) Zbl 0137.11602
[a16] H.P. McKean Jr., "Brownian motion with a several-dimensional time" Theory Probab. Appl. , 8 (1963) pp. 335–354
[a17] G.M. Molchan, "Some problems for Lévy's Brownian motion" Theory Probab. Appl. , 12 (1967) pp. 682–690
[a18] S. Orey, W. Pruitt, "Sample functions of the N-parameter Wiener process" Ann. of Probab. , 1 (1973) pp. 138–163 MR0346925 Zbl 0284.60036
[a19] S.R. Paranjape, C. Park, "Distribution of the supremum of the two-parameter Yeh–Wiener process on the boundary" J. Appl. Probab. , 10 (1973) pp. 875–880 MR0381015 Zbl 0281.60081
[a20] S.R. Paranjape, C. Park, "Laws of iterated logarithm of multiparameter Wiener process" J. Multivariate Anal. , 3 (1973) pp. 132–136 MR0326852
[a21] V.I. Piterbarg, "Asymptotic methods in the theory of Gaussian processes and fields" , Amer. Math. Soc. (1996) MR1361884 Zbl 0841.60024
[a22] Yu.A. Rosanov, "Markov random fields" , Springer (1982)
[a23] J.B. Walsh, "Propagation of singularities in the Brownian sheet" Ann. of Probab. , Ann. 10 (1982) pp. 279–288 MR0647504 Zbl 0528.60076
[a24] M.I. Yadrenko, "Spectral theory of random fields" , Optim. Software (1983) MR0697386 Zbl 0539.60048
How to Cite This Entry:
Wiener field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wiener_field&oldid=50353
This article was adapted from an original article by M.I. Yadrenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article