On the first passage times of spatial branching processes

Given a discrete-time non-lattice supercritical branching random walk (or branching Brownian motion) in $\mathbb{R}^d$, we investigate its first passage time to a shifted unit ball of a distance $x$ from the origin, conditioned upon survival. We provide precise asymptotics up to $O(1)$ (tightness) for the first passage time as a function of $x$ as $x\to\infty$. Our proof for the branching random walk case employs a careful multi-scale analysis on the genealogy of particles within a distance of $\asymp\log x$ near extrema of a one-dimensional branching random walk, where the cluster structure plays a crucial role.

This is based on joint work with Jose Blanchet, Wei Cai, and Shaswat Mohanty.