ASYMPTOTIC BEHAVIORS OF SUBCRITICAL BRANCHING KILLED BROWNIAN MOTION WITH DRIFT

In this paper, we study asymptotic behaviors of a subcritical branching Brownian motion with drift −ρ, killed upon exiting (0, ∞), and offspring distribution {p_k: k ≥ 0}. Let ζ̴^−ρ be the extinction time of this subcritical branching killed Brownian motion, M̴_t^−ρ the maximal position of all the particles alive at time t and M̴−^ρ:= max_t≥0 M̴_t⁻ρ the all-time maximal position. Let P_x 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)p_k < ∞, we establish the decay rates of P_x(ζ̴^−ρ > t) and P_x(M̴−^ρ > y) as t and y respectively tend to ∞. We also establish the decay rate of P_x(M̴_t^−ρ > z(t, ρ)) as t → ∞, where z(t, ρ) = ^√tz − ρt for ρ ≤ 0 and z(t, ρ) = z for ρ > 0. As a consequence, we obtain a Yaglom-type limit theorem.