Abstract.
We investigate the random continuous trees called Lévy trees, which are obtained as scaling limits of discrete Galton-Watson trees. We give a mathematically precise definition of these random trees as random variables taking values in the set of equivalence classes of compact rooted ℝ-trees, which is equipped with the Gromov-Hausdorff distance. To construct Lévy trees, we make use of the coding by the height process which was studied in detail in previous work. We then investigate various probabilistic properties of Lévy trees. In particular we establish a branching property analogous to the well-known property for Galton-Watson trees: Conditionally given the tree below level a, the subtrees originating from that level are distributed as the atoms of a Poisson point measure whose intensity involves a local time measure supported on the vertices at distance a from the root. We study regularity properties of local times in the space variable, and prove that the support of local time is the full level set, except for certain exceptional values of a corresponding to local extinctions. We also compute several fractal dimensions of Lévy trees, including Hausdorff and packing dimensions, in terms of lower and upper indices for the branching mechanism function ψ which characterizes the distribution of the tree. We finally discuss some applications to super-Brownian motion with a general branching mechanism.
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References
Aldous, D.: The continuum random tree I. Ann. Probab. 19, 1–28 (1991)
Aldous, D.: The continuum random tree III. Ann. Probab. 21, 248–289 (1993)
Aldous, D., Miermont, G., Pitman, J.: The exploration process of inhomogeneous continuum random trees, and an extension of Jeulin’s local time identity. Probab. Theory Relat. Fields 129, 182–218 (2004)
Dawson, D.A.: Measure-valued Markov processes. Ecole d’été de probabilités de Saint-Flour 1991. Lecture Notes in Math. Springer, Berlin, 1541, 1–260 (1993)
Dawson, D.A., Perkins, E.A.: Historical Processes. Memoirs Amer. Math. Soc. 454, 1991
Dress, A., Moulton, V., Terhalle, W.: T-theory: An overview. Eur. J. Combinatorics 17, 161–175 (1996)
Delmas, J.F.: Path properties of superprocesses with a general branching mechanism. Ann. Probab. 27, 1099–1134 (1999)
Duquesne, T.: A limit theorem for the contour process of conditioned Galton-Watson trees. Ann. Probab. 31, 996–1027 (2003)
Duquesne, T., Le Gall, J.F.: (2002) Random Trees, Lévy Processes and Spatial Branching Processes. Astérisque, 281
Duquesne, T., Le Gall, J.F.: (2004) The Hausdorff measure of stable trees. In preparation
Dynkin, E.B., Kuznetsov, S.E.: (2003) ℕ-measures for branching exit Markov systems and their applications to differential equations. To appear Probab. Theory Relat. Fields
El Karoui, N., Roelly, S.: Propriétés de martingales, explosion et représentation de Lévy-Khintchine d’une classe de processus de branchement à valeurs mesures. Stoch. Process. Appl. 38, 239–266 (1991)
Evans, S.N., Pitman, J.W., Winter, A.: Rayleigh processes, real trees and root growth with re-grafting. Preprint 2003
Falconer, K.J.: (2003) Fractal Geometry: Mathematical Foundations and Applications. 2nd ed. Wiley, New York
Gromov, M.: Metric Structures for Riemannian and Non-Riemannian Spaces. Progress in Mathematics. Birkhäuser, Boston, 1999
Haas, B., Miermont, G.: The genealogy of self-similar fragmentations with negative index as a continuum random tree. Preprint, 2003
Lamperti, J.: Continuous-state branching processes. Bull. Am. Math. Soc. 73, 382–386 (1967)
Ledoux, M., Talagrand, M.: Probability in Banach Spaces. Springer, Berlin, 1991
Le Gall, J.F.: Spatial Branching Processes, Random Snakes and Partial Differential Equations. Lectures in Mathematics ETH Zürich. Birkhäuser, Boston, 1999
Le Gall, J.F., Le Jan, Y.: Branching processes in Lévy processes: The exploration process. Ann. Probab. 26, 213–252 (1998)
Le Gall, J.F., Le Jan, Y.: Branching processes in Lévy processes: Laplace functionals of snakes and superprocesses. Ann. Probab. 26, 1407–1432 (1998)
Le Gall, J.F., Perkins, E.A.: The Hausdorff measure of the support of two-dimensional super-Brownian motion. Ann. Probab. 23, 1719–1747 (1995)
Limic, V.: A LIFO queue in heavy traffic. Ann. Appl. Probab. 11, 301–331 (2001)
Mattila, P.: (1995) Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, Cambridge
Miermont, G.: Self-similar fragmentations derived from the stable tree: Spliting at heights. Probab. Theory Relat. Fields 127, 423–454 (2003)
Paulin, F.: The Gromov topology on ℝ-trees. Topology Appl. 32, 197–221 (1989)
Perkins, E.A.: A space-time property of a class of measure-valued branching diffusions. Trans. Amer. Math. Soc. 305, 743–795 (1988)
Perkins, E.A.: Polar sets and multiple points for super-Brownian motion. Ann. Probab. 18, 453–491 (1990)
Serlet, L.: Some dimension results for super-Brownian motion. Probab. Theory Relat. Fields 101, 371–391 (1995)
Tribe, R.: Path properties of superprocesses. Ph.D. Thesis, University of British Columbia, 1989
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Duquesne, T., Gall, JF. Probabilistic and fractal aspects of Lévy trees. Probab. Theory Relat. Fields 131, 553–603 (2005). https://doi.org/10.1007/s00440-004-0385-4
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DOI: https://doi.org/10.1007/s00440-004-0385-4