On the volume of the shrinking branching Brownian sausage

TL;DR

This paper investigates the asymptotic behavior of the volume of a shrinking branching Brownian sausage in multiple dimensions, providing almost sure limit theorems for large times based on advanced probabilistic techniques.

Contribution

It introduces a detailed analysis of the volume of a shrinking BBM-sausage, extending previous work by deriving almost sure limit theorems in all dimensions.

Findings

Almost sure convergence of the volume as time tends to infinity

Asymptotic formulas for the volume in all dimensions

Connection to classical Wiener sausage and Brownian hitting probabilities

Abstract

The branching Brownian sausage in $\mathbb{R}^d$ was defined by Engl\"ander in [Stoch. Proc. Appl. 88 (2000)] similarly to the classical Wiener sausage, as the random subset of $\mathbb{R}^d$ scooped out by moving balls of fixed radius with centers following the trajectories of the particles of a branching Brownian motion (BBM). We consider a $d$-dimensional dyadic BBM, and study the large-time asymptotic behavior of the volume of the associated exponentially shrinking branching Brownian sausage (BBM-sausage). Using a previous result on the density of the support of BBM, and some well-known results on the classical Wiener sausage and Brownian hitting probabilities, we obtain almost sure limit theorems as time tends to infinity on the volume of the shrinking BBM-sausage in all dimensions.