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Branching Brownian motion seen from its tip. (English) Zbl 1284.60154
Authors’ abstract: It has been conjectured since the work of S. P. Lalley and T. Sellke [Ann. Probab. 15, 1052–1061 (1987; Zbl 0622.60085)] that branching Brownian motion seen from its tip (e.g. from its rightmost particle) converges to an invariant point process. Very recently, it emerged that this can be proved in several different ways (see, e.g. [É. Brunet and B. Derrida, J. Stat. Phys. 143, No. 3, 420–446 (2011; Zbl 1219.82095); L.-P. Arguin et al., Ann. Appl. Probab. 22, No. 4, 1693–1711 (2012; Zbl 1255.60152); Probab. Theory Relat. Fields 157, No. 3–4, 535–574 (2013; Zbl 1286.60045)])). The structure of this extremal point process turns out to be a Poisson point process with exponential intensity in which each atom has been decorated by an independent copy of an auxiliary point process. The main goal of the present work is to give a complete description of the limit object via an explicit construction of this decoration point process. Another proof and description has been obtained independently by Arguin et al. [Zbl 1255.60152].

MSC:
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60G70 Extreme value theory; extremal stochastic processes
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[1] Aïdékon, E.: Convergence in law of the minimum of a branching random walk. Ann. Probab. ArXiv 1101.1810 [math.PR] (to appear) · Zbl 0326.60093
[2] Arguin, L.-P., Bovier, A., Kistler, N.: The genealogy of extremal particles of branching Brownian motion. ArXiv 1008.4386 [math.PR] (2010) · Zbl 1236.60081
[3] Arguin, L.-P., Bovier, A., Kistler, N.: Poissonian statistics in the extremal process of branching Brownian motion. ArXiv 1010.2376 [math.PR] (2010) · Zbl 1255.60152
[4] Arguin, L.-P., Bovier, A., Kistler, N.: The extremal process of branching Brownian motion. ArXiv 1103.2322 [math.PR] (2011) · Zbl 1236.60081
[5] Bramson, M, Maximal displacement of branching Brownian motion, Commun. Pure Appl. Math., 31, 531-581, (1978) · Zbl 0361.60052
[6] Bramson, M.: Convergence of solutions of the Kolmogorov equation to travelling waves. Mem. Am. Math. Soc. 44(285) (1983) · Zbl 0517.60083
[7] Brunet, E; Derrida, B, Statistics at the tip of a branching random walk and the delay of traveling waves, Eur. Phys. Lett., 87, 60010, (2009)
[8] Brunet, E., Derrida, B.: A branching random walk seen from the tip. ArXiv cond-mat/1011.4864 (2010)
[9] Chauvin, B; Rouault, A, Étude de l’équation KPP et du branchement brownien en zones sous-critique et critique. application aux arbres spatiaux, C. R Acad. Sci. Paris Sér. I Math., 304, 19-22, (1987) · Zbl 0602.60075
[10] Chauvin, B; Rouault, A, KPP equation and supercritical branching Brownian motion in the subcritical speed area. application to spatial trees, Probab. Theory Relat. Fields, 80, 299-314, (1988) · Zbl 0653.60077
[11] Chauvin, B.: Multiplicative martingales and stopping lines for branching Brownian motion. Ann. Probab. 30:1195-1205 (1991) · Zbl 0738.60079
[12] Denisov, IV, A random walk and a Wiener process near a maximum, Theory Probab. Appl., 28, 821-824, (1983) · Zbl 0544.60070
[13] Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs (1964) · Zbl 0144.34903
[14] Hardy, R., Harris, S.C.: A new formulation of the spine approach to branching diffusions. arXiv: 0611.054 [math.PR] (2006)
[15] Harris, SC, Travelling-waves for the FKPP equation via probabilistic arguments, Proc. R. Soc. Edinb., 129A, 503-517, (1999) · Zbl 0946.35040
[16] Harris, JW; Harris, SC; Kyprianou, AE, Further probabilistic analysis of the Fisher-Kolmogorov-Petrovskii-Piscounov equation: one-sided traveling waves, Ann. Inst. H. Poincaré Probab. Stat., 42, 125-145, (2006) · Zbl 1093.60059
[17] Harris, S.C., Roberts, M.I.: The many-to-few lemma and multiple spines. arXiv:1106.4761 [math.PR] (2011) · Zbl 0544.60070
[18] Ikeda, N; Nagasawa, M; Watanabe, S, On branching Markov processes, Proc. Japan Acad., 41, 816-821, (1965) · Zbl 0224.60038
[19] Imhof, J-P, Density factorizations for Brownian motion, meander and the three-dimensional Bessel process, and applications, J. Appl. Probab., 21, 500-510, (1984) · Zbl 0547.60081
[20] Jagers, P, General branching processes as Markov fields, Stoch. Process. Appl., 32, 193-212, (1989) · Zbl 0678.92009
[21] Kallenberg, O.: Random measures. Springer, Berlin (1983) · Zbl 0544.60053
[22] Kyprianou, A, Travelling wave solutions to the K-P-P equation: alternatives to Simon harris’ probabilistic analysis, Ann. Inst. H. Poincaré Probab. Statist., 40, 53-72, (2004) · Zbl 1042.60057
[23] Lalley, SP; Sellke, T, A conditional limit theorem for frontier of a branching Brownian motion, Ann. Probab., 15, 1052-1061, (1987) · Zbl 0622.60085
[24] McKean, HP, Application of Brownian motion to the equation of Kolmogorov-Petrovskii-piskunov, Commun. Pure Appl. Math., 28, 323-331, (1975) · Zbl 0316.35053
[25] Madaule, T.: Convergence in law for the branching random walk seen from its tip. arXiv:1107.2543 [math.PR] (2011)
[26] Maillard, P.: A characterisation of superposable random measures. arXiv:1102.1888 [math.PR] (2011)
[27] Maillard, P.: Branching Brownian motion with selection of the \(N\) right-mostparticles: an approximate model. arXiv:1112.0266v2 [math.PR] (2012)
[28] Neveu, J.: Multiplicative martingales for spatial branching processes. Seminar on Stochastic Processes, Princeton, pp 223-242. Progr. Probab. Statist., vol. 15, Birkhäuser, Boston (1988) · Zbl 0652.60089
[29] Resnick, S.I.: Extreme values, regular variation, and point processes. Applied Probability Series, vol. 4, Springer, Berlin (1987) · Zbl 0633.60001
[30] Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin (1999) · Zbl 0917.60006
[31] Williams, D, Path decomposition and continuity of local time for one-dimensional diffusions. I, Proc. Lond. Math. Soc. (3), 28, 738-768, (1974) · Zbl 0326.60093
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