2019/11/25 by Öz, Mehmet
#60D05 #60F15 #60J80 #92D25 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1911.10919
The branching Brownian sausage in ℝd was defined by Engländer in [Stoch. Proc. Appl. 88 (2000)] similarly to the classical Wiener sausage, as the random subset of ℝd 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.