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

TL;DR

This paper investigates the geometric and probabilistic properties of the Brownian map and super-Brownian motion, establishing asymptotic relations for connected components and local times using advanced stochastic process techniques.

Contribution

It provides new asymptotic results linking the structure of the Brownian map with local times of super-Brownian motion through the Brownian snake framework.

Findings

r^3N(h,r) converges to a constant times the density at h

Asymptotics for vertices at height h with descendants at h+r

Approximation of super-Brownian local times by upcrossing numbers

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.