Asymptotic Expansions for Additive Measures of Branching Brownian Motions

Let N(t) be the collection of particles alive at time t in a branching Brownian motion in \(\mathbb {R}^d\), and for \(u\in N(t)\), let \({\textbf{X}}_u(t)\) be the position of particle u at time t. For \(\theta \in \mathbb {R}^d\), we define the additive measures of the branching Brownian motion by

$$\begin{aligned}{} & {} \mu _t^\theta (\textrm{d}{\textbf{x}}):= e^{-(1+\frac{\Vert \theta \Vert ^2}{2})t}\sum _{u\in N(t)} e^{-\theta \cdot {\textbf{X}}_u(t)} \delta _{\left( {\textbf{X}}_u(t)+\theta t\right) }(\textrm{d}{\textbf{x}}),\\{} & {} \quad \textrm{here}\,\, \Vert \theta \Vert \mathrm {is\, the\, Euclidean\, norm\, of}\,\, \theta . \end{aligned}$$

In this paper, under some conditions on the offspring distribution, we give asymptotic expansions of arbitrary order for \(\mu _t^\theta (({\textbf{a}}, {\textbf{b}}])\) and \(\mu _t^\theta ((-\infty , {\textbf{a}}])\) for \(\theta \in \mathbb {R}^d\) with \(\Vert \theta \Vert <\sqrt{2}\), where \((\textbf{a}, \textbf{b}]:=(a_1, b_1]\times \cdots \times (a_d, b_d]\) and \((-\infty , \textbf{a}]:=(-\infty , a_1]\times \cdots \times (-\infty , a_d]\) for \(\textbf{a}=(a_1,\cdots , a_d)\) and \(\textbf{b}=(b_1,\cdots , b_d)\). These expansions sharpen the asymptotic results of Asmussen and Kaplan (Stoch Process Appl 4(1):1–13, 1976) and Kang (J Korean Math Soc 36(1): 139–157, 1999) and are analogs of the expansions in Gao and Liu (Sci China Math 64(12):2759–2774, 2021) and Révész et al. (J Appl Probab 42(4):1081–1094, 2005) for branching Wiener processes (a particular class of branching random walks) corresponding to \(\theta ={\textbf{0}}\).