ar X iv : 1 61 1 . 01 83 3 v 1 [ m at h . PR ] 6 N ov 2 01 6 Number of particles absorbed in a BBM on the extinction event

نویسنده

  • Pierre-Antoine Corre
چکیده

We consider a branching Brownian motion which starts from 0 with drift μ ∈ R and we focus on the number Zx of particles killed at −x, where x > 0. Let us call μ0 the critical drift such that there is a positive probability of survival if and only if μ > −μ0. Maillard [17] and Berestycki et al. [5] have study Zx in the case μ ≤ −μ0 and μ ≥ μ0 respectively. We complete the picture by considering the case where μ > −μ0 on the extinction event. More precisely we study the asymptotic of qi(x) := P (Zx = i, ζx < ∞). We show that the radius of convergence R(μ) of the corresponding power series increases as μ increases, up until μ = μc ∈ [−μ0,+∞] after which it is constant. We also give a necessary and sufficient condition for μc < +∞. In addition, finer asymptotics are also obtained, which highlight three different regimes depending on μ < μc, μ = μc or μ > μc.

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تاریخ انتشار 2016