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
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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:
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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 |
Wiener field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wiener_field&oldid=24596