2023/08/24 by Amine Asselah, Asselah, Amine, Izumi Okada +4 · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · #Diffusion and Search Dynamics #FOS: Mathematics #Point processes and geometric inequalities #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2308.12948
openalex publication_date 2023/08/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We show that the range of a critical branching random walk conditioned to survive forever and the Minkowski sum of two independent simple random walk ranges are intersection-equivalent in any dimension d≥ 5, in the sense that they hit any finite set with comparable probability, as their common starting point is sufficiently far away from the set to be hit. Furthermore, we extend a discrete version of Kesten, Spitzer and Whitman's result on the law of large numbers for the volume of a Wiener sausage. Here, the sausage is made of the Minkowski sum of N independent simple random walk ranges in ℤd, with d>2N, and of a finite set A⊂ ℤd. When properly normalised the volume of the sausage converges to a quantity equivalent to the capacity of A with respect to the kernel K(x,y)=(1+‖x-y‖)2N-d. As a consequence, we establish a new relation between capacity and \it branching capacity.