Asymptotic behaviors of subcritical branching killed Brownian motion with drift

In this paper, we study asymptotic behaviors of a subcritical branching killed Brownian motion with drift -ρ and offspring distribution ₖ:k≥ 0\. Let ζ-ρ be the extinction time of this subcritical branching killed Brownian motion, Mₜ-ρ the maximal position of all the particles alive at time t and M-ρ:=maxt≥ 0Mₜ-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(ζ-ρ>t) and Pₓ(M-ρ>y) as t and y tend to ∞ respectively. We also establish the decay rate of Pₓ(Mₜ-ρ>z(t,ρ)) as t→∞, where z(t,ρ)=√tz-ρ t for ρ≤ 0 and z(t,ρ)=z for ρ>0. As a consequence, we obtain a Yaglom-type limit theorem.