A Tree-valued Markov Process Associated with an Admissible Family of Branching Mechanisms
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摘要
Let E?R be an interval. By studying an admissible family of branching mechanisms{ψt,t ∈E} introduced in Li [Ann. Probab., 42, 41-79(2014)], we construct a decreasing Levy-CRT-valued process {Tt, t ∈ E} by pruning Lévy trees accordingly such that for each t ∈E, Tt is a ψt-Lévy tree. We also obtain an analogous process {Tt*,t ∈E} by pruning a critical Levy tree conditioned to be infinite. Under a regular condition on the admissible family of branching mechanisms, we show that the law of {Tt,t ∈E} at the ascension time A := inf{t ∈E;Tt is finite} can be represented by{Tt*,t∈E}.The results generalize those studied in Abraham and Delmas [Ann. Probab., 40, 1167-1211(2012)].
Let E?R be an interval. By studying an admissible family of branching mechanisms{ψt,t ∈E} introduced in Li [Ann. Probab., 42, 41-79(2014)], we construct a decreasing Levy-CRT-valued process {Tt, t ∈ E} by pruning Lévy trees accordingly such that for each t ∈E, Tt is a ψt-Lévy tree. We also obtain an analogous process {Tt*,t ∈E} by pruning a critical Levy tree conditioned to be infinite. Under a regular condition on the admissible family of branching mechanisms, we show that the law of {Tt,t ∈E} at the ascension time A := inf{t ∈E;Tt is finite} can be represented by{Tt*,t∈E}.The results generalize those studied in Abraham and Delmas [Ann. Probab., 40, 1167-1211(2012)].
Let E?R be an interval. By studying an admissible family of branching mechanisms{ψt,t ∈E} introduced in Li [Ann. Probab., 42, 41-79(2014)], we construct a decreasing Levy-CRT-valued process {Tt, t ∈ E} by pruning Lévy trees accordingly such that for each t ∈E, Tt is a ψt-Lévy tree. We also obtain an analogous process {Tt*,t ∈E} by pruning a critical Levy tree conditioned to be infinite. Under a regular condition on the admissible family of branching mechanisms, we show that the law of {Tt,t ∈E} at the ascension time A := inf{t ∈E;Tt is finite} can be represented by{Tt*,t∈E}.The results generalize those studied in Abraham and Delmas [Ann. Probab., 40, 1167-1211(2012)].
引文
[1] Abraham, R., Delmas, J. F.:A continuum-tree-valued Markov process. Ann. Probab., 40, 1167–1211(2012)
[2] Abraham, R., Delmas, J. F.:The forest associated with the record process on Lévy tree. Stochastic Process.Appl., 123(9), 3497–3517(2013)
[3] Abraham, R., Delmas, J. F., He, H.:Pruning Galton–Watson trees and tree-valued Markov processes.Ann. Inst. H. Poincaré Probab. Statist., 48, 688–705(2012)
[4] Abraham, R., Delmas, J. F., He, H.:Pruning of CRT-sub-trees. Stochastic Process. Appl., 125(4), 1569–1604(2012)
[5] Abraham, R., Delmas, J. F., Hoscheit, P.:A note on Gromov–Hausdorff–Prokhorov distance between(locally)compact measure spaces. Electron. J. Probab., 18(14), 1–21(2013)
[6] Abraham, R., Delmas, J. F., Hoscheit, P.:Exit times for an increasing Lévy tree-valued process. Probab.Theory Related Fields, 159(1–2), 357–405(2014)
[7] Abraham, R., Delmas, J. F., Voisin, G.:Pruning a Lévy continuum random tree. Electron. J. Probab., 15,1429–1473(2010)
[8] Aldous, D.:The continuum random tree I. Ann. Probab., 19, 1–28(1991)
[9] Aldous, D.:The continuum random tree III. Ann. Probab., 21, 248–289(1993)
[10] Aldous, D., Pitman, J.:Tree-valued Markov chains derived from Galton–Watson processes. Ann. Inst. H.Poincaré Probab. Statist., 34, 637–686(1998)
[11] Duquesne, T., Le Gall, J. F.:Random Trees, Lévy Processes and Spatial Branching Processes, Astérisque,2002
[12] Duquesne, T., Le Gall, J. F.:Probabilistic and fractal aspects of Lévy trees. Probab. Theory Related Fields,131, 553–603(2005)
[13] Duquesne, T., Winkel, M.:Growth of Lévy trees. Probab. Theory Related Fields, 139, 313–371(2007)
[14] Evans, S.:Probability and real trees, volume 1920 of Ecole d’été de Probabilités de Saint-Flour, Lecture Notes in Math, Springer, 2008
[15] Greven, A., Pfaffelhuber, P., Winter, A.:Convergence in distribution of random metric measure spaces(Λ-coalescent measure trees). Probab. Theory Related Fields, 145, 285–322(2008)
[16] Hoscheit, P.:Processusa valeurs dans les arbres alé atoires continus, Le Grade De Docteur En Sciences De L’Université Paris-Est, 2012
[17] Le Gall, J. F.:Random trees and applications. Probab. Surv., 2, 245–311(2005)
[18] Le Gall, J. F., Le Jan, Y.:Branching processes in Lévy processes:the exploration process. Ann. Probab.,26, 213–252(1998)
[19] Le Gall, J. F., Le Jan, Y.:Branching processes in Lévy processes:Laplace functionals of snakes and superprocesses. Ann. Probab., 26, 1407–1432(1998)
[20] Li, Z.:Path-valued branching processes and nonlocal branching superprocesses. Ann. Probab., 42, 41–79(2014)
[2] Abraham, R., Delmas, J. F.:The forest associated with the record process on Lévy tree. Stochastic Process.Appl., 123(9), 3497–3517(2013)
[3] Abraham, R., Delmas, J. F., He, H.:Pruning Galton–Watson trees and tree-valued Markov processes.Ann. Inst. H. Poincaré Probab. Statist., 48, 688–705(2012)
[4] Abraham, R., Delmas, J. F., He, H.:Pruning of CRT-sub-trees. Stochastic Process. Appl., 125(4), 1569–1604(2012)
[5] Abraham, R., Delmas, J. F., Hoscheit, P.:A note on Gromov–Hausdorff–Prokhorov distance between(locally)compact measure spaces. Electron. J. Probab., 18(14), 1–21(2013)
[6] Abraham, R., Delmas, J. F., Hoscheit, P.:Exit times for an increasing Lévy tree-valued process. Probab.Theory Related Fields, 159(1–2), 357–405(2014)
[7] Abraham, R., Delmas, J. F., Voisin, G.:Pruning a Lévy continuum random tree. Electron. J. Probab., 15,1429–1473(2010)
[8] Aldous, D.:The continuum random tree I. Ann. Probab., 19, 1–28(1991)
[9] Aldous, D.:The continuum random tree III. Ann. Probab., 21, 248–289(1993)
[10] Aldous, D., Pitman, J.:Tree-valued Markov chains derived from Galton–Watson processes. Ann. Inst. H.Poincaré Probab. Statist., 34, 637–686(1998)
[11] Duquesne, T., Le Gall, J. F.:Random Trees, Lévy Processes and Spatial Branching Processes, Astérisque,2002
[12] Duquesne, T., Le Gall, J. F.:Probabilistic and fractal aspects of Lévy trees. Probab. Theory Related Fields,131, 553–603(2005)
[13] Duquesne, T., Winkel, M.:Growth of Lévy trees. Probab. Theory Related Fields, 139, 313–371(2007)
[14] Evans, S.:Probability and real trees, volume 1920 of Ecole d’été de Probabilités de Saint-Flour, Lecture Notes in Math, Springer, 2008
[15] Greven, A., Pfaffelhuber, P., Winter, A.:Convergence in distribution of random metric measure spaces(Λ-coalescent measure trees). Probab. Theory Related Fields, 145, 285–322(2008)
[16] Hoscheit, P.:Processusa valeurs dans les arbres alé atoires continus, Le Grade De Docteur En Sciences De L’Université Paris-Est, 2012
[17] Le Gall, J. F.:Random trees and applications. Probab. Surv., 2, 245–311(2005)
[18] Le Gall, J. F., Le Jan, Y.:Branching processes in Lévy processes:the exploration process. Ann. Probab.,26, 213–252(1998)
[19] Le Gall, J. F., Le Jan, Y.:Branching processes in Lévy processes:Laplace functionals of snakes and superprocesses. Ann. Probab., 26, 1407–1432(1998)
[20] Li, Z.:Path-valued branching processes and nonlocal branching superprocesses. Ann. Probab., 42, 41–79(2014)