Growth of Lévy trees

Probability Theory and Related Fields

Volumn 139, Issue 3-4, 2007, Pages 313-371

  Duquesne, Thomas ^a   Winkel, Matthias ^b  

^a UNIVERSITÉ PARIS SACLAY   (France)

^b UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

Continuous state branching process; Continuum random tree; Edge lengths; Galton Watson branching process; Genealogy; Gromov Hausdorff topology; Percolation; Real tree; Tree valued Markov process

Indexed keywords

EID: 34547190353     PISSN: 01788051     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00440-007-0064-3     Document Type: Article

References (35)

1. Abraham, R.: Un arbre aléatoire infini associé à l'excursion Brownienne. In: Sém. de Proba., vol. XXVI, pp. 374-397. Springer, Berlin (1992)
2. Aldous D. and Pitman J. (1998). Tree-valued Markov chains derived from Galton-Watson processes. Ann. Inst. H. Poincaré 34: 637-686
3. Aldous D.J. (1991). The continuum random tree I. Ann. Probab. 19: 1-28
4. Athreya, K., Ney, P.: Branching process. Number 196 in Grundlehren der Mathematischen Wissenschaften. Springer, Berlin (1972)
5. Bingham N.H. (1976). Continuous branching processes and spectral positivity. Stoch. Process. Appl. 4: 217-242
6. Burago, Y., Burago, D., Ivanov, S.: A Course in Metric Geometry, vol. 33. AMS, Boston (2001)
7. Dress A. (1984). Trees, tight extensions of metric spaces and the cohomological dimension of certain groups: a note on combinatorial properties of metric spaces. Adv. Math. 53: 321-402
8. Dress A., Moulton V. and Terhalle W. (1996). T-theory: an overview. Eur. J. Comb. 17: 161-175
9. Dress A. and Terhalle W. (1996). The real tree. Adv. Math. 120: 283-301
10. Duquesne, T., Le Gall, J.-F.: Random Trees, Lévy Processes and Spatial Branching Processes. Astérisque no 281 (2002)
11. Duquesne T. and Le Gall J.-F. (2005). Probabilistic and fractal aspects of Lévy trees. Probab. Theory Relat. Fields 131(4): 553-603
12. Dynkin, E.B.: A type of interaction between superprocesses and branching particle systems. In: Measure-valued processes, stochastic partial differential equations, and interacting systems (Montreal, PQ, 1992). CRM Proc. Lecture Notes, vol. 5, pp. 83-86. Amer. Math. Soc., Providence (1994)
13. Evans S. (2000). Snakes and spiders: Brownian motion on real trees. Probab. Theory Relat. Fields 117(3): 361-386
14. Evans S.N. and Winter A. (2006). Subtree prune and regraft: a reversible real tree-valued Markov process. Ann. Probab. 34(3): 918-961
15. Evans S.N. and O'Connell N. (1994). Weighted occupation time for branching particle systems and a representation for the supercritical superprocess. Can. Math. Bull. 37(2): 187-196
16. Evans S.N., Pitman J. and Winter A. (2006). Rayleigh processes, real trees, and root growth with re-grafting. Probab. Theory Relat. Fields 134(1): 81-126
17. Feller, W.: An Introduction to Probability Theory and Its Applications, vol. II, 2nd edn. Wiley, New York (1971)
18. Geiger J. and Kauffmann L. (2004). The shape of large Galton-Watson trees with possibly infinite variance. Rand. Struct. Alg. 25(3): 311-335
19. Gromov, M.: Metric Structures for Riemannian and non-Riemannian Spaces. Progress in Mathematics. Birkhäuser, Basel (1999)
20. Hambly, B.M., Lyons, T.J.: Uniqueness for the signature of a path of bounded variation and continuous analogues for the free group. Preprint (2004)
21. Hobson, D.G.: Marked excursions and random trees. In: Lecture Notes in Mathematics. Séminaire de Probabilités XXXIV, vol. 1729, pp. 289-301. Springer, Berlin (2000)
22. Jirina M. (1958). Stochastic branching processes with continous state-space. Czech. Math. J. 8: 292-313
23. Lamperti J. (1967). Continuous-state branching processes. Bull. Amer. Math. Soc. 73: 382-386
24. Lamperti J. (1967). The limit of a sequence of branching processes. Z. Wahrsch. Verw. Gebiete 7: 271-288
25. Lamperti, J.: Limiting distributions of branching processes. In: 5th Berkeley Symposium, vol. II, Part 2, pp. 225-241 (1967)
26. Le Gall J.-F. and Le Jan Y. (1998). Branching processes in Lévy processes: the exploration process. Ann. Probab. 26(1): 213-252
27. Le Gall J.-F. and Le Jan Y. (1999). Branching processes in Lévy processes: Laplace functionals of snakes and superprocesses. Ann. Probab. 26: 1407-1432
28. Le Jan Y. (1991). Superprocesses and projective limits of branching Markov processes. Ann. Inst. H. Poincaré 27: 91-106
29. Lyons, R., Peres, Y: Probability on Trees and Networks. Cambridge University Press, Cambridge, in progress. Current version published on the web at http://php.indiana.edu/~rdlyons (2004)
30. Neveu J. (1986). Arbres et processus de Galton-Watson. Ann. Inst. H. Poincaré 26: 199-207
31. Paulin F. (1988). Topologie de Gromov équivariante, structures hyperboliques et arbres réels. Invent. Math. 94(1): 53-80
32. Paulin F. (1989). The Gromov topology on real-trees. Topology Appl. 32(3): 197-221
33. Pitman, J.: Combinatorial stochastic processes, Ecole d'Eté de Probabilités de Saint-Flour XXXII, 2002. Lecture Notes in Mathematics, vol. 1875. Springer, Berlin (2006)
34. Pitman J. and Winkel M. (2005). Growth of the Brownian forest. Ann. Probab. 33(6): 2188-2211
35. Salminen P. (1992). Cutting Markovian trees. Ann. Acad. Scient. Fenn. A. I. Math. 17: 123-137