Asymptotic expansion for additive measure of branching Brownian motion

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 ${\mathbf{X}}_u(t)$ 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$${\mu }_t^{\theta }(\mathrm{d}\mathbf{x}):=e^{-(1+\frac{\mathrm{\parallel }\theta {\mathrm{\parallel }}^2}{2})t}\sum _{u\in N(t)}{e}^{-\theta \cdot {\mathbf{X}}_u(t)}{\delta }_{({\mathbf{X}}_u(t)+\theta t)}(\mathrm{d}\mathbf{x})\mathrm{.}$$ In this paper, under some conditions on the offspring distribution, we give asymptotic expansions of arbitrary order for ${\mu }_t^{\theta }((\mathbf{a},\mathbf{b}])$ and ${\mu }_t^{\theta }((-\mathrm{\infty },\mathbf{a}])$ for $\theta \in {\mathbb{R}}^d$ with $\mathrm{\parallel }\theta \mathrm{\parallel }<\sqrt{2}$. These expansions sharpen the asymptotic results of Asmussen and Kaplan (1976) and Kang (1999), and are analogs of the expansions in Gao and Liu (2021) and Révész, Rosen and Shi (2005) for branching Wiener processes (a particular class of branching random walks) corresponding to $\theta =\mathbf{0}$.