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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.