The Brownian cactus II. Upcrossings and local times of super-Brownian motion

Abstract

We study properties of the random metric space called the Brownian map. For every h>0, we consider the connected components of the complement of the open ball of radius h centered at the root, and we let N(h,r) be the number of those connected components that intersect the complement of the ball of radius h+r. We then prove that r^3N(h,r) converges as r tends to 0 to a constant times the density at h of the profile of distances from the root. In terms of the Brownian cactus, this gives asymptotics for the number of vertices at height h that have descendants at height h+r. Our proofs are based on a similar approximation result for local times of super-Brownian motion by upcrossing numbers. Our arguments make a heavy use of the Brownian snake and its special Markov property.