Probabilistic and fractal aspects of Lévy trees. (English) Zbl 1070.60076

Duquesne, Thomas; Le Gall, Jean-François

Probab. Theory Relat. Fields 131, No. 4, 553-603 (2005).

The random continuous trees called Lévy trees are obtained as scaling limits of discrete Galton-Watson trees. One of the authors’ goals is to initiate a probabilistic theory of \({\mathbb R}\)-trees, by starting with the fundamental case of Lévy trees. The authors give a mathematically precise definition of these random trees as random variables taking values in the set \({\mathcal T}_H\) of equivalence classes of compact rooted \({\mathbb R}\)-trees, equipped with the Gromov-Hausdorff distance \(d_H\). To construct such Lévy trees, they take advantage of the coding by the height process \(H=\) \(( H_t; t \geq 0)\) which was studied in detail in their previous work [“Random trees, Lévy processes and spatial branching processes” (2002; Zbl 1037.60074)].
The claim is here that the sample path of the height process \(H\) under the excursion measure \(N\) codes a random continuous tree called the \(\psi\)-Lévy tree with the branching mechanism \(\psi\) defined on \([0, \infty)\). More precisely, \(\psi\) is a nonnegative function and of the form \[ \psi(\lambda) = \alpha \lambda + \beta \lambda^2 + \int_{(0, \infty)} ( e^{- \lambda r} - 1 + \lambda r) \pi(dr) \] where \(\lambda \geq 0\), \(\alpha \geq 0\), \(\beta \geq 0\) and \(\pi\) is a \(\sigma\)-finite measure on \((0, \infty)\) such that \(\int_{(0, \infty)} ( r \wedge r^2) \pi (dr)\) \(< \infty\). The condition \(\int_1^{\infty} \psi(u)^{-1} du\) \(< \infty\) is assumed, which is equivalent to the a.s. extinction of the continuous-state branching process with \(\psi\). This is nothing but a necessary condition for the compactness of the associated genealogical tree. The most important cases are, of course, the quadratic branching case \(\psi(\lambda)\) \(=\) \(c \lambda^2\) and the stable case \(\psi(\lambda)\) \(=\) \(c \lambda^{\gamma}\) with \(1 < \gamma < 2\).
Various probabilistic properties of Lévy trees are investigated. For example, a branching property of the Lévy tree is established, which is analogous to the well-known property for Galton-Watson trees: namely, conditionally given the tree below level \(a\) \((> 0)\), the subtrees originating from that level are distributed as the atoms of a Poisson point measure whose intensity involves a local time measure \(\ell^a(d\sigma)\) supported on the vertical at distance \(a\) from the root (Theorem 4.2). The authors show some regularity properties of local times \(( \ell^a )\) in the space variable, asserting that there exists a modification of the collection \(( \ell^a\); \(a \geq 0)\) in such a way that the mapping \(a \mapsto \ell^a\) is \(\Theta(d{\mathcal T})\) a.e. càdlàg for weak topology on finite measure on \({\mathcal T}\) (Theorem 4.3), where \({\mathcal T}\) is an \({\mathbb R}\)-tree and \(\Theta(d{\mathcal T})\) is the law of the Lévy tree. And also they prove that for every \(a \in {\mathcal E}\), the topological support of \(\ell^a\) is \({\mathcal T}(a) \setminus \{ \sigma_a \}\) (Theorem 4.4), where \({\mathcal T}(a)\) is the level set, \(\sigma_a\) is the unique extinction point at level \(a\), and \({\mathcal E}\) is the set of all extinction levels, asserting that the support of local time \(\ell^a\) is the full level set, except for certain exceptional values of \(a\) corresponding to local extinctions. Moreover, several fractal dimensions of Lévy trees, including Hausdorff and packing dimensions, are also computed in terms of lower and upper indices for the branching mechanism function \(\psi\) which characterizes the distribution of the tree (Theorem 5.5).
Finally some applications to super-Brownian motion with a general branching mechanism are discussed as well. For other related works, see e.g. D. Aldous, G. Miermont and J. Pitman [Probab. Theory Relat. Fields 129, No. 2, 182–218 (2004; Zbl 1056.60011)], where a different class of continuous random trees obtained as weak limits of birthday trees, instead of the Galton-Watson trees considered here are discussed.

Reviewer: Isamu Dôku (Saitama)

MSC:

60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60J55	Local time and additive functionals
60G17	Sample path properties

Keywords:

Galton-Watson tree; height process; branching property; local time measure; topological support; fractal dimension

Citations:

Zbl 1037.60074; Zbl 1056.60011

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