Difference between revisions of "Wiener field"
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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" />. | 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" />. | ||
− | 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 | + | 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]]]). |
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 | 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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====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <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 17:02, 15 April 2012
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 |
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Wiener field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wiener_field&oldid=13757