Preprint:
On the explosion of the local times of Brownian sheet
along lines
D. Khoshnevisan, P. Révész, and Z. Shi
Abstract.
One can view a 2-parameter Brownian sheet
{ W(s,t); 0 \le s,t } as a stream of interacting
Brownian motions { W(s,.); 0 \le s }. Given this
view point, we aim to continue the analysis of
Walsh (1978) on the local times of the stream
W(s,.), when s \ge 0.
Oue main result is a kind of maximal
inequality that, in particular, verifies the following
conjecture of Khoshnevisan (1995): as s goes to 0,
the local times of W(s,.) explode almost surely.
Two other applications of this maximal inequality are presented,
one to a capacity estimate in classical Wiener space, and
one to a uniform ratio ergodic theorem in Wiener space. The latter
readily implies a quasi-sure ergodic theorem. We also
present a sharp Hölder condition
for the local times of the mentioned Brownian streams
that refines earlier results of (Lacey 1990; Révész 1985;
Walsh 1978).
Keywords.
Brownian sheet; local times along lines; ergodic theorems
AMS Classification (2000)
Primary. 60G60;
Secondary. 60K35, 60B12, 60G17
Support. Research supported in part by grants from
- U.S. National Science Foundation (D. Kh.)
- North Atlantic Treaty Organization (D. Kh. and Z. S.)
- Hungarian National Foundation for Scientific Research (P. R.)
Pre/E-Prints. This paper is available in
Davar Khoshnevisan
Department of Mathematics
University of Utah
155 S, 1400 E JWB 233
Salt Lake City, UT 84112-0090, U.S.A.
davar@math.utah.edu
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Zhan Shi
Laboratoire de Probabilités
Université Paris VI
4 Place Jussieu
F-75252 Paris, Cedex 05
France
zhan@proba.jussieu.fr
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Pál Révész
Institut für Statistik
und Wahrscheinlichkeitstheorie
Technische Universität Wien
Wiedner Haupstrasse 8-10/1071
A-1040, Wien, Austria
revesz@ci.tuwien.ac.at
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Last Update: January 24, 2001
© 2001 - Davar Khoshnevisan, P. Révész, and Z. Shi