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 μ _t^θ (dx):= e^-(1+‖θ‖ ^2/2)t∑ _u∈ N(t) e^-θ·X_u(t)δ _( X_u(t)+θ t) (dx), 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 μ _t^θ ((a, b]) and μ _t^θ ((-∞ , a]) for θ∈ℝ^d with ‖θ‖ <√(2) , where (a, b]:=(a_1, b_1]×⋯× (a_d, b_d] and (-∞ , a]:=(-∞ , a_1]×⋯× (-∞ , a_d] for a=(a_1,⋯ , a_d) and b=(b_1,⋯ , 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 .