Asymptotic expansion for branching killed Brownian motion with drift

TL;DR

This paper derives detailed asymptotic expansions for the distribution of particles in a branching Brownian motion with drift, killed at zero, extending previous results and providing sharper estimates under specific conditions.

Contribution

It introduces new asymptotic expansion formulas for particle distributions in branching Brownian motion with drift, improving upon earlier work by Louidor, Saglietti, and Kesten.

Findings

Extended asymptotic expansions for particle counts in branching Brownian motion.

Sharpened estimates under the condition $ heta \\in [0,\\sqrt{2})$ and specific tail assumptions.

Generalization of previous asymptotic results with broader applicability.

Abstract

Let $Z_t^{(0,\infty)}$ be the point process formed by the positions of all particles alive at time $t$ in a branching Brownian motion with drift and killed upon reaching 0. We study the asymptotic expansions of $Z_t^{(0,\infty)}(A)$ for $A= (a,b)$ and $A=(a,\infty)$ under the assumption that $\sum_{k=1}^\infty k(\log k)^{1+\lambda} p_k <\infty$ for large $\lambda$ in the regime of $\theta \in [0,\sqrt{2})$. These results extend and sharpen the results of Louidor and Saglietti [J. Stat. Phys, 2020] and Kesten [Stochastic Process. Appl., 1978].