2024/07/01 by Hou, Haojie, Ren, Yan-Xia, Song, Renming +1
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2407.01816
In this paper, we study asymptotic behaviors of a subcritical branching killed Brownian motion with drift -ρ and offspring distribution \pk:k≥ 0\. Let \widetildeζ-ρ be the extinction time of this subcritical branching killed Brownian motion, \widetildeMt-ρ the maximal position of all the particles alive at time t and \widetildeM-ρ:=maxt≥ 0\widetildeMt-ρ the all time maximal position. Let ℙx be the law of this subcritical branching killed Brownian motion when the initial particle is located at x∈ (0,∞). Under the assumption ∑k=1^∞ k (log k) pk <∞, we establish the decay rates of ℙx(\widetildeζ-ρ>t) and ℙx(\widetildeM-ρ>y) as t and y tend to ∞ respectively. We also establish the decay rate of ℙx(\widetildeMt-ρ>z(t,ρ)) as t→∞, where z(t,ρ)=√(t)z-ρt for ρ≤ 0 and z(t,ρ)=z for ρ>0. As a consequence, we obtain a Yaglom-type limit theorem.