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

Endre Csáki, Davar Khoshnevisan and Zhan
Shi

Summary: Our main intention is to describe the
behavior of the (cumulative) distribution function of
the random variable M_{0,1}:=
sup_{0< s,t< 1}W(s,t)
near 0, where W denotes one-dimensional,
two-parameter Brownian sheet. A remarkable result of
Florit and Nualart (1995) 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.

Keywords: Tail probability, quasi-sure analysis,
Brownian sheet.

AMS 1991 subject classification: 60G60; 60G17.