Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 ``Probabilités et Modèles Aléatoires''

### Boundary crossings and the distribution function of the maximum of Brownian sheet

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