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 $$\mathbb {R}^d$$ R d , and for $$u\in N(t)$$ u ∈ N ( t ) , let $${\textbf{X}}_u(t)$$ X u ( t ) be the position of particle u at time t. For $$\theta \in \mathbb {R}^d$$ θ ∈ R d , we define the additive measures of the branching Brownian motion by $$\begin{aligned}{} & {} \mu _t^\theta (\textrm{d}{\textbf{x}}):= e^{-(1+\frac{\Vert \theta \Vert ^2}{2})t}\sum _{u\in N(t)} e^{-\theta \cdot {\textbf{X}}_u(t)} \delta _{\left( {\textbf{X}}_u(t)+\theta t\right) }(\textrm{d}{\textbf{x}}),\\{} & {} \quad \textrm{here}\,\, \Vert \theta \Vert \mathrm {is\, the\, Euclidean\, norm\, of}\,\, \theta . \end{aligned}$$ μ t θ ( d x ) : = e - ( 1 + ‖ θ ‖ 2 2 ) t ∑ u ∈ N ( t ) e - θ · X u ( t ) δ X u ( t ) + θ t ( d x ) , here ‖ θ ‖ is the Euclidean norm of θ . In this paper, under some conditions on the offspring distribution, we give asymptotic expansions of arbitrary order for $$\mu _t^\theta (({\textbf{a}}, {\textbf{b}}])$$ μ t θ ( ( a , b ] ) and $$\mu _t^\theta ((-\infty , {\textbf{a}}])$$ μ t θ ( ( - ∞ , a ] ) for $$\theta \in \mathbb {R}^d$$ θ ∈ R d with $$\Vert \theta \Vert