Levy Trees Research Articles

A Tree-valued Markov Process Associated with an Admissible Family of Branching Mechanisms

Let $$\mathfrak{T}$$ ⊂ ℝ be an interval. By studying an admissible family of branching mechanisms {ψt, t ∈ $$\mathfrak{T}$$ } introduced in Li [Ann. Probab., 42, 41–79 (2014)], we construct a decreasing Levy-CRT-valued process $$\left\{\mathfrak{T}_t,t\in\mathfrak{T}\right\}$$ by pruning Levy trees accordingly such that for each t ∈ $$\mathfrak{T}$$ , $$\mathcal{T}_t$$ is a ψt-Levy tree. We also obtain an analogous process $$\left\{\mathfrak{T}_t^*,t\in\mathfrak{T}\right\}$$ by pruning a critical Levy tree conditioned to be infinite. Under a regular condition on the admissible family of branching mechanisms, we show that the law of $$\left\{\mathfrak{T}_t,t\in\mathfrak{T}\right\}$$ at the ascension time A:= inf{t ∈ $$\mathfrak{T}$$ ; $$\mathcal{T}_t$$ is finite} can be represented by $$\left\{\mathfrak{T}_t^*,t\in\mathfrak{T}\right\}$$ . The results generalize those studied in Abraham and Delmas [Ann. Probab., 40, 1167–1211 (2012)].

Scaling Limit of a Limit Order Book Model via the Regenerative Characterization of Lévy Trees

We consider the following Markovian dynamic on point processes: at constant rate and with equal probability, either the rightmost atom of the current configuration is removed, or a new atom is added at a random distance from the rightmost atom. Interpreting atoms as limit buy orders, this process was introduced by Lakner et al. to model a one-sided limit order book. We consider this model in the regime where the total number of orders converges to a reflected Brownian motion, and complement the results of Lakner et al. by showing that, in the case where the mean displacement at which a new order is added is positive, the measure-valued process describing the whole limit order book converges to a simple functional of this reflected Brownian motion. Our results make it possible to derive useful and explicit approximations on various quantities of interest such as the depth or the total value of the book. Our approach leverages an unexpected connection with Levy trees. More precisely, the cornerstone of our approach is the regenerative characterization of Levy trees due to Weill, which provides an elegant proof strategy which we unfold.

Subordination of trees and the Brownian map

We discuss subordination of random compact \({\mathbb R}\)-trees. We focus on the case of the Brownian tree, where the subordination function is given by the past maximum process of Brownian motion indexed by the tree. In that particular case, the subordinate tree is identified as a stable Levy tree with index 3/2. As a more precise alternative formulation, we show that the maximum process of the Brownian snake is a time change of the height process coding the Levy tree. We then apply our results to properties of the Brownian map. In particular, we recover, in a more precise form, a recent result of Miller and Sheffield identifying the metric net associated with the Brownian map.

Scaling limits for a family of unrooted trees

We introduce weights on the unrooted unlabelled plane trees as follows: for each p ≥ 1, let μp be a probability measure on the set of nonnegative integers whose mean is bounded by 1; then the μp-weight of a plane tree t is defined as Π μ_p(degree(v) − 1), where the product is over the set of vertices v of t. We study the random plane tree T_p which has a fixed diameter p and is sampled according to probabilities proportional to these μ_p-weights. We prove that, under the assumption that the sequence of laws μ_p, p ≥ 1, belongs to the domain of attraction of an infinitely divisible law, the scaling limits of (T_p , p ≥ 1) are random compact real trees called the unrooted Levy trees, which have been introduced in Duquesne and Wang (2016+).

Decomposition of Lévy trees along their diameter

We study the diameter of Levy trees that are random compact metric spaces obtained as the scaling limits of Galton–Watson trees. Levy trees have been introduced by Le Gall & Le Jan (Ann. Probab. 26 (1998) 213–252) and they generalise Aldous’ Continuum Random Tree (1991) that corresponds to the Brownian case. We first characterize the law of the diameter of Levy trees and we prove that it is realized by a unique pair of points. We prove that the law of Levy trees conditioned to have a fixed diameter r ∈ (0, ∞) is obtained by glueing at their respective roots two independent size-biased Levy trees conditioned to have height r/2 and then by uniformly re-rooting the resulting tree; we also describe by a Poisson point measure the law of the subtrees that are grafted on the diameter. As an application of this decomposition of Levy trees according to their diameter, we characterize the joint law of the height and the diameter of stable Levy trees conditioned by their total mass; we also provide asymptotic expansions of the law of the height and of the diameter of such normalised stable trees, which generalises the identity due to Szekeres (In Combinatorial Mathematics, X (Adelaide, 1982) (1983) 392–397 Springer) in the Brownian case.

A note on the scaling limits of contour functions of Galton-Watson trees

Recently, Abraham and Delmas constructed the distributions of super-critical Levy trees truncated at a fixed height by connecting super-critical Levy trees to (sub)critical Levy trees via a martingale transformation. A similar relationship also holds for discrete Galton-Watson trees. In this work, using the existing works on the convergence of contour functions of (sub)critical trees, we prove that the contour functions of truncated super critical Galton-Watson trees converge weakly to the distributions constructed by Abraham and Delmas.

On the re-rooting invariance property of Lévy trees

We prove a strong form of the invariance under re-rooting of the distribution of the continuous random trees called Levy trees. This expends previous results due to several authors.

Regenerative real trees

In this work, we give a description of all σ-finite measures on the space of rooted compact R-trees which satisfy a certain regenerative property. We show that any infinite measure which satisfies the regenerative property is the of a Levy tree, that is, the of a tree-valued random variable that describes the genealogy of a population evolving according to a continuous-state branching process. On the other hand, we prove that a probability measure with the regenerative property must be the law of the genealogical tree associated with a continuous-time discrete-state branching process.

Growth of Lévy trees

We construct random locally compact real trees called Levy trees that are the genealogical trees associated with continuous-state branching processes. More precisely, we define a growing family of discrete Galton–Watson trees with i.i.d. exponential branch lengths that is consistent under Bernoulli percolation on leaves; we define the Levy tree as the limit of this growing family with respect to the Gromov–Hausdorff topology on metric spaces. This elementary approach notably includes supercritical trees and does not make use of the height process introduced by Le Gall and Le Jan to code the genealogy of (sub)critical continuous-state branching processes. We construct the mass measure of Levy trees and we give a decomposition along the ancestral subtree of a Poisson sampling directed by the mass measure.

Probabilistic and fractal aspects of Lévy trees

We investigate the random continuous trees called Levy 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 Levy 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 Levy 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 Levy 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.