Branching capacity and Brownian snake capacity

TL;DR

This paper establishes that the rescaled branching capacity of symmetric branching random walks converges to the Brownian snake capacity in high dimensions, providing a link between discrete and continuous stochastic processes.

Contribution

It introduces the Brownian snake capacity in Euclidean space and proves its convergence as a scaling limit of the branching capacity, with a focus on convergence rates.

Findings

Rescaled branching capacity converges to Brownian snake capacity

Established precise convergence rates for hitting probability approximations

Linked discrete branching processes with continuous Brownian snake models

Abstract

The branching capacity has been introduced by [Zhu 2016] as the limit of the hitting probability of a symmetric branching random walk in $\mathbb Z^d$, $d\ge 5$. Similarly, we define the Brownian snake capacity in $\mathbb R^d$, as the scaling limit of the hitting probability by the Brownian snake starting from afar. Then, we prove our main result on the vague convergence of the rescaled branching capacity towards this Brownian snake capacity. Our proof relies on a precise convergence rate for the approximation of the branching capacity by hitting probabilities.