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 R^d, and for u∈N(t), let X_u(t) be the position of particle u at time t. For θ∈R^d, we define the additive measures of the branching Brownian motion by (Formula presented.) In this paper, under some conditions on the offspring distribution, we give asymptotic expansions of arbitrary order for μ_t^θ((a,b]) and μ_t^θ((-∞,a]) for θ∈R^d with ‖θ‖<2, where (a,b]:=(a₁,b₁]×⋯×(a_d,b_d] and (-∞,a]:=(-∞,a₁]×⋯×(-∞,a_d] for a=(a₁,⋯,a_d) and b=(b₁,⋯,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 θ=0.