Probabilistic and fractal aspects of Lévy trees

By Thomas Duquesne, Jean-François Le Gall. 2005

268 citations recorded by the source corpus.

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.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2ba83cbf16675ff11355b5f0ed65a37d7c55b57b

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Citations in the corpus

Random trees and applications (2005, methodology): “If in addition D is regular and g is continuous, then u solves the problem ∆u = 4u2 in D u|∂D = g (36) where the notation u|∂D = g means that for every y ∈ ∂D, lim D∋x→y u(x) = g(y) .”
The topological structure of scaling limits of large planar maps (2006, background): “Therefore we may and will sometimes view Z as indexed by vertices of the tree Te.”
Sub-Gaussian tail bounds for the width and height of conditioned Galton--Watson trees (2010, background): “This result has been extended to sequences of random trees that converge to Lévy trees; see [14, 15].”
Conditioned Brownian trees (2005, unknown): “Roughly speaking, N0 is obtained by the same construction as above, but instead of considering a normalized Brownian excursion, we now let e be distributed according to the (infinite) Itô measure of Brownian excursions.”
Random stable looptrees (2013, unknown): “Fix ε > 0 and let ( t ( ε ) i ) 1 ≤ i ≤ N ε be an increasing enumeration of the elements of the ﬁnite set { t ∈ [0 , 1]; ∆ t > ε 1 /α } and set t ( ε ) 0 = 0 and t ( ε ) where by deﬁnition We shall now prove that, for every η ∈ (0 , 1 /α ) , This will entail that a.s. dim H ( L α ) < α (1 + η ) /…”
Geodesics in large planar maps and in the Brownian map (2008, background): “1 of [14], the metric space (Tg, dg) is a (compact) real tree.”
Scaling Limits of Random Graphs from Subcritical Classes (2014, background): “d Tn;; (d) ! (T e;d e;r 0(T e)) in the metric space (K;d GH). This invariance principle is due to Aldous [5] and there exist various extensions. See for example Duquesne [20], Duquesne and Le Gall [21], Haas and Miermont [25]. Adopting common terminology, we will often write in the sequel ˙ 2 p n T”
Scaling limits of random trees and planar maps (2011, background): “The remaining part of the argument is straightforward: See Section 2 in [19].”
Random real trees (2006, methodology): “The remaining part of the argument is straightforward: See Section 2 in [22] (or Lemma 3.”
Scaling limits of Markov branching trees with applications to Galton–Watson and random unordered trees (2010, result): “In particular, we recover the results of Aldous [4] and Duquesne [18] showing that the so-called Brownian and stable trees [2, 19, 20, 29] are universal limits for conditioned Galton–Watson trees.”
Random geometry on the sphere (2014, background): “[25]), which by definition is rooted at ρg = pg(0) = pg(1).”
BETA-COALESCENTS AND CONTINUOUS STABLE RANDOM TREES (2006, background): “7 in [18]) that if ∆Za > 0 then a is necessarily a level where exactly one excursion is splitting into infinitely many smaller ones (i.”
Scaling limits of random planar maps with a unique large face (2012, background): “Note that the limit Te is quite different from the Brownian map mentioned above; it is a (random) compact tree, and thus contractible, that is, of the same homotopy type as a point, and its Hausdorff dimension is 2 [24, 28].”
Rayleigh processes, real trees, and root growth with re-grafting (2004, result): “Some of the results on the space of R-trees obtained in this paper have already been found useful in the study scaling limits of Galton‐Watson branching processes in [ DLG04 ].”
Invariance principles for spatial multitype Galton–Watson trees (2006, methodology): “This limiting forest is made of tree components that are described by a Poisson process whose intensity measure is the σ -finite Brownian continuum tree measure Θ of [5].”