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Asymptotic Behaviors of Subcritical Branching Killed Brownian Motion with Drift

arxiv(2024)

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