Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 60G60 Random fields
- 60G17 Sample path properties

**Résumé:** Our main intention is to describe the
behavior of the (cumulative) distribution
function of the random variable $M_{0,1}
:= \sup_{0\le s,t\le 1} W(s,t)$ near $0$,
where $W$ denotes one-dimensional,
two-parameter Brownian sheet. A remarkable
result of Florit and Nualart asserts that
$M_{0,1}$ has a smooth density function
with respect to Lebesgue's measure. Our
estimates, in turn, seem to imply that the
behavior of the density function of
$M_{0,1}$ near 0 is quite exotic and, in
particular, there is no clear-cut notion
of a two-parameter reflection principle.
We also consider the supremum of Brownian
sheet over rectangles that are away from
the origin. We apply our estimates to get
an infinite dimensional analogue of
Hirsch's theorem for Brownian motion.

**Mots Clés:** *Tail probability ; quasi-sure analysis ; Brownian sheet*

**Date:** 2000-04-21

**Prépublication numéro:** *PMA-587*