On the Distribution of Bubbles of the Brownian Sheet

Abstract

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 $\lbrack 0, t\rbrack^2$, whose maximum height is greater than $h$. Evidently, $\lim_{h\downarrow 0} N(t; h) = \infty$ and $\lim_{t\uparrow\infty} N(t; h) = \infty$. It is the goal of this paper to provide fairly accurate estimates on $N(t; h)$ both as $t\rightarrow\infty$ and as $h\rightarrow 0$. Loosely speaking, we show that there are of order $h^{-3}$ many such bubbles as $h \downarrow 0$ and $t^3$ many, as $t\uparrow \infty$.