zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
The contour of splitting trees is a Lévy process. (English) Zbl 1190.60083
Author’s abstract: Splitting trees are those random trees where individuals give birth at a constant rate during a lifetime with general distribution, to i.i.d. copies of themselves. The width process of a splitting tree is then a binary, homogeneous Crump-Mode-Jagers (CMJ) process, and is not Markovian unless the lifetime distribution is exponential (or a Dirac mass at ${\infty }$). Here, we allow the birth rate to be infinite, that is, pairs of birth times and life spans of newborns form a Poisson point process along the lifetime of their mother, with possibly infinite intensity measure. A splitting tree is a random (so-called) chronological tree. Each element of a chronological tree is a (so-called) existence point $(v, \tau )$ of some individual $v$ (vertex) in a discrete tree where $\tau $ is a nonnegative real number called chronological level (time). We introduce a total order on existence points, called linear order, and a mapping $\varphi $ from the tree into the real line which preserves this order. The inverse of $\varphi $ is called the exploration process, and the projection of this inverse on chronological levels the contour process. For splitting trees truncated up to level $\tau $, we prove that a thus defined contour process is a Lévy process reflected below $\tau $ and killed upon hitting 0. This allows one to derive properties of (i) splitting trees: conceptual proof of Le Gall-Le Jan’s theorem in the finite variation case, exceptional points, coalescent point process and age distribution; (ii) CMJ processes: one-dimensional marginals, conditionings, limit theorems and asymptotic numbers of individuals with infinite versus finite descendances.

60J80Branching processes
37E25Maps of trees and graphs
60G51Processes with independent increments; Lévy processes
60G55Point processes
60G70Extreme value theory; extremal processes (probability theory)
60J55Local time, additive functionals
60J75Jump processes
60J85Applications of branching processes
92D25Population dynamics (general)
Full Text: DOI
[1] Aldous, D. (1991). The continuum random tree. I. Ann. Probab. 19 1-28. · Zbl 0722.60013 · doi:10.1214/aop/1176990534
[2] Aldous, D. (1993). The continuum random tree. III. Ann. Probab. 21 248-289. · Zbl 0791.60009 · doi:10.1214/aop/1176989404
[3] Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121 . Cambridge Univ. Press, Cambridge. · Zbl 0861.60003
[4] Bertoin, J., Fontbona, J. and Martínez, S. (2008). On prolific individuals in a supercritical continuous-state branching process. J. Appl. Probab. 45 714-726. · Zbl 1154.60066 · doi:10.1239/jap/1222441825
[5] Bertoin, J. and Le Gall, J.-F. (2000). The Bolthausen-Sznitman coalescent and the genealogy of continuous-state branching processes. Probab. Theory Related Fields 117 249-266. · Zbl 0963.60086 · doi:10.1007/s004400000053
[6] Dress, A. W. M. and Terhalle, W. F. (1996). The real tree. Adv. Math. 120 283-301. · Zbl 0855.05040 · doi:10.1006/aima.1996.0040
[7] Duquesne, T. (2007). The coding of compact real trees by real valued functions. Preprint. Available at arXiv PR/0604106. · Zbl 1183.94031
[8] Duquesne, T. and Le Gall, J. F. (2002). Random trees, Lévy processes and spatial branching processes. Astérisque 281 1-147. · Zbl 1037.60074
[9] Evans, S. N. (2009). Probability and Real Trees. Lecture Notes in Math. 1920 . Springer, Berlin.
[10] Geiger, J. and Kersting, G. (1997). Depth-first search of random trees, and Poisson point processes. In Classical and Modern Branching Processes ( Minneapolis , MN , 1994). IMA Math. Appl. 84 111-126. Springer, New York. · Zbl 0867.60061
[11] Jaffard, S. (1999). The multifractal nature of Lévy processes. Probab. Theory Related Fields 114 207-227. · Zbl 0947.60039 · doi:10.1007/s004400050224
[12] Jiřina, M. (1958). Stochastic branching processes with continuous state space. Czechoslovak Math. J. 8 292-313. · Zbl 0168.38602 · eudml:11934
[13] Lambert, A. (2002). The genealogy of continuous-state branching processes with immigration. Probab. Theory Related Fields 122 42-70. · Zbl 1020.60074 · doi:10.1007/s004400100155
[14] Lambert, A. (2003). Coalescence times for the branching process. Adv. in Appl. Probab. 35 1071-1089. · Zbl 1040.60073 · doi:10.1239/aap/1067436335
[15] Lambert, A. (2008). Population dynamics and random genealogies. Stoch. Models 24 45-163. · Zbl 05383029
[16] Lambert, A. (2009). The allelic partition for coalescent point processes. Markov Processes Relat. Fields . Preprint. To appear. Available at · Zbl 1177.92026
[17] Lambert, A. (2010). Spine decompositions of Lévy trees. To appear. In preparation.
[18] Le Gall, J.-F. (1993). The uniform random tree in a Brownian excursion. Probab. Theory Related Fields 96 369-383. · Zbl 0794.60080 · doi:10.1007/BF01292678
[19] Le Gall, J.-F. (2005). Random trees and applications. Probab. Surv. 2 245-311 (electronic). · Zbl 1189.60161 · doi:10.1214/154957805100000140 · eudml:229216
[20] Le Gall, J.-F. and Le Jan, Y. (1998). Branching processes in Lévy processes: The exploration process. Ann. Probab. 26 213-252. · Zbl 0948.60071 · doi:10.1214/aop/1022855417
[21] Neveu, J. (1986). Arbres et processus de Galton-Watson. Ann. Inst. H. Poincaré Probab. Statist. 22 199-207. · Zbl 0601.60082 · numdam:AIHPB_1986__22_2_199_0 · eudml:77276
[22] Nerman, O. (1981). On the convergence of supercritical general (C-M-J) branching processes. Z. Wahrsch. Verw. Gebiete 57 365-395. · Zbl 0451.60078 · doi:10.1007/BF00534830
[23] O’Connell, N. (1995). The genealogy of branching processes and the age of our most recent common ancestor. Adv. in Appl. Probab. 27 418-442. · Zbl 0837.60080 · doi:10.2307/1427834
[24] Popovic, L. (2004). Asymptotic genealogy of a critical branching process. Ann. Appl. Probab. 14 2120-2148. · Zbl 1062.92048 · doi:10.1214/105051604000000486 · euclid:aoap/1099674091
[25] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion , 3rd ed. Grundlehren der Mathematischen Wissenschaften [ Fundamental Principles of Mathematical Sciences ] 293 . Springer, Berlin. · Zbl 0917.60006
[26] Taïb, Z. (1992). Branching Processes and Neutral Evolution. Lecture Notes in Biomathematics 93 . Springer, Berlin. · Zbl 0748.60081