NDLI: Probabilistic and fractal aspects of Lévy trees

Content Provider	Springer Nature Link
Author	Duquesne, Thomas Gall, Jean François Le
Copyright Year	2004
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.
Starting Page	553
Ending Page	603
Page Count	51
File Format	PDF
ISSN	01788051
Journal	Probability Theory and Related Fields
Volume Number	131
Issue Number	4
e-ISSN	14322064
Language	English
Publisher	Springer-Verlag
Publisher Date	2004-11-11
Publisher Place	Berlin/Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Mathematical Biology in General Quantitative Finance Probability Theory and Stochastic Processes Mathematical and Computational Physics Statistics for Business/Economics/Mathematical Finance/Insurance Operation Research/Decision Theory
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability Analysis Statistics, Probability and Uncertainty

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