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

Our main intention is to describe the behavior of the (cumulative) distribu tion function of the random variable M-0,M-1:=sup(0 less than or equal tos, t less than or equal to1) W(s,t) near 0, where W denotes one-dimensional, t wo-parameter Brownian sheet. A remarkable result of Florit and Nualart asse rts that M-0,M-1 has a smooth density function with respect to Lebesgue's m easure (cf. Florit and Nualart, 1995. Statist. Probab. Lett. 22, 25-31). Ou r estimates, in turn, seem to imply that the behavior of the density functi on of M-0,M-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 su premum of Brownian sheet over rectangles that are away from the origin. We apply our estimates to get an infinite-dimensional analogue of Hirsch's the orem for Brownian motion. (C) 2000 Elsevier Science B.V. All rights reserve d.