Wiener field

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