# The Brownian cactus II: upcrossings and local times of super-Brownian motion

Research paper by Jean-François Le Gall

Indexed on: 04 Jul '14Published on: 04 Jul '14Published in: Probability Theory and Related Fields

#### Abstract

We study properties of the random metric space called the Brownian map. For every $$r>0$$, we consider the connected components of the complement of the open ball of radius $$r$$ centered at the root, and we let $$\mathbf {N}_{r,\varepsilon }$$ be the number of those connected components that intersect the complement of the ball of radius $$r+\varepsilon$$. We then prove that $$\varepsilon ^3\mathbf {N}_{r,\varepsilon }$$ converges as $$\varepsilon \rightarrow 0$$ to a constant times the density at $$r$$ of the profile of distances from the root. In terms of the Brownian cactus, this gives asymptotics for the number of points at height $$r$$ that have descendants at height $$r+\varepsilon$$. 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.