856 research outputs found
Fractal properties of the random string processes
Let be a random string taking values
in , specified by the following stochastic partial differential
equation [Funaki (1983)]: where is
an -valued space-time white noise. Mueller and Tribe (2002)
have proved necessary and sufficient conditions for the -valued
process to hit points and to have double
points. In this paper, we continue their research by determining the Hausdorff
and packing dimensions of the level sets and the sets of double times of the
random string process . We also consider
the Hausdorff and packing dimensions of the range and graph of the string.Comment: Published at http://dx.doi.org/10.1214/074921706000000806 in the IMS
Lecture Notes Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
Joint continuity of the local times of fractional Brownian sheets
Let be an -fractional Brownian
sheet with index defined by
where
are independent copies of a real-valued fractional Brownian
sheet . We prove that if , then the
local times of are jointly continuous. This verifies a conjecture of Xiao
and Zhang (Probab. Theory Related Fields 124 (2002)). We also establish sharp
local and global H\"{o}lder conditions for the local times of . These
results are applied to study analytic and geometric properties of the sample
paths of .Comment: Published in at http://dx.doi.org/10.1214/07-AIHP131 the Annales de
l'Institut Henri Poincar\'e - Probabilit\'es et Statistiques
(http://www.imstat.org/aihp/) by the Institute of Mathematical Statistics
(http://www.imstat.org
Local times of multifractional Brownian sheets
Denote by a function in
with values in . Let
be an
-multifractional Brownian sheet (mfBs) with Hurst functional .
Under some regularity conditions on the function , we prove the
existence, joint continuity and the H\"{o}lder regularity of the local times of
. We also determine the Hausdorff dimensions of the level sets
of . Our results extend the corresponding results for
fractional Brownian sheets and multifractional Brownian motion to
multifractional Brownian sheets.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ126 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Critical Brownian sheet does not have double points
We derive a decoupling formula for the Brownian sheet which has the following
ready consequence: An -parameter Brownian sheet in has double
points if and only if . In particular, in the critical case where ,
the Brownian sheet does not have double points. This answers an old problem in
the folklore of the subject. We also discuss some of the geometric consequences
of the mentioned decoupling, and establish a partial result concerning
-multiple points in the critical case .Comment: Published in at http://dx.doi.org/10.1214/11-AOP665 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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