Probabilistic and fractal aspects of Levy trees

Simple item page
dc.creatorDuquesne, Thomas
dc.creatorGall, Jean-Francois Le
dc.date2005-01-06
dc.date.accessioned2026-07-07T05:15:52Z
dc.date.available2026-07-07T05:15:52Z
dc.descriptionWe 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.
dc.description48 pages
dc.identifierhttps://arxiv.org/abs/math/0501079
dc.identifierhttp://arxiv.org/abs/math/0501079
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/73779
dc.subjectProbability
dc.subject60J80,60G57
dc.titleProbabilistic and fractal aspects of Levy trees
dc.typetext

Files

Collections

arXiv