Wiener field

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

-parameter Wiener field (Brownian motion).

Let be a Gaussian separable real-valued field on with zero mean and covariance function

where . Such a field can be regarded as the distribution function of a white noise on , which is a random function on bounded Borel sets in such that has a normal distribution with zero mean and covariance function [a11]. Here, denotes the Lebesgue measure on . The following equality holds: , where is a parallelepiped in .

The random field 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 [a3]. For any fixed time variables, is a one-parameter Wiener process as a function of the free time variable. Some properties of are similar to the corresponding properties of the Wiener process: the sample functions of almost surely satisfy Hölder's stochastic condition with exponent [a2]; various forms of the law of the iterated logarithm hold true ([a5], [a18], [a20]). An exact formula for exists only for the Wiener process. For , only lower and upper bounds (, [a12]) and some asymptotic formulas for [a21] have been derived so far (1998). The level sets of 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 .

The Wiener process is a Markov process: conditional of the present value , the past () and the future () are independent. For the multivariate case there are several definitions of the Markovian property. Let denote a family of Jordan surfaces in . Each such surface divides into two parts: , the interior of , or the "past" , and , the exterior of , or the "future" . A random field is said to be Markovian with respect to the family if for arbitrary from and arbitrary , , the random variables and are conditionally independent given [a24]. A Wiener field is a Markovian field with respect to the family consisting of all finite unions of rectangles whose sides are parallel to the coordinate axes ([a22], [a23]). For , its sharp field and germ field are defined, respectively, by and , where is an -neighbourhood of . A Wiener sheet is germ Markovian, i.e. for every bounded subset , the fields and are conditionally independent given ([a6], [a7], [a22]).

Among the objects closely related to are the Wiener pillow and the Wiener bridge. These are Gaussian random fields (cf. also Random field) on with zero mean and covariance functions

respectively.

Lévy -parameter Brownian motion.

This is a Gaussian random field on with zero mean and covariance function

where [a15]. When , becomes a Wiener process. The random variables clearly form a Wiener process if moves along some semi-straight line with terminal point . has the following representation in terms of white noise:

where is the surface area of the -dimensional unit sphere [a17]. H.P. McKean Jr. [a16] has shown that is germ Markovian with respect to closed bounded subsets in for each odd , whereas for each even 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