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

Our main intention is to describe the behavior of the (cumulative) distribution function of the random variable M0,1 ≔ sup0⩽s,t⩽1 W(s,t) near 0, where W denotes one-dimensional, two-parameter Brownian sheet. A remarkable result of Florit and Nualart asserts that M0,1 has a smooth density function with respect to Lebesgueʹs measure (cf. Florit and Nualart, 1995. Statist. Probab. Lett. 22, 25–31). Our estimates, in turn, seem to imply that the behavior of the density function of M0,1 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 Hirschʹs theorem for Brownian motion.