ON THE DISTRIBUTION OF BUBBLES OF THE BROWNIAN SHEET

Let W be a real-valued, two-parameter Brownian sheet. Let us define N( t; h) to be the total number of bubbles of W in [0, t](2), whose maxim um height is greater than h. Evidently, lim(h down arrow 0) N(t; h) = infinity and lim(t) (up arrow) (infinity) N(t; h) = infinity. It is th e goal of this paper to provide fairly accurate estimates on N(t; h) b oth as t --> infinity and as h --> 0. Loosely speaking, we show that t here are of order h(-3) many such bubbles as h down arrow 0 and t(3) m any, as t up arrow infinity.