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 (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 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 . [ABSTRACT FROM AUTHOR]