Random Trees, Lévy Processes and Spatial Branching Processes

By Thomas Duquesne, Jean-François Le Gall. Astérisque, 2005

339 citations recorded by the source corpus.

We investigate the genealogical structure of general critical or subcritical continuous-state branching processes. Analogously to the coding of a discrete tree by its contour function, this genealogical structure is coded by a real-valued stochastic process called the height process, which is itself constructed as a local time functional of a Levy process with no negative jumps. We present a detailed study of the height process and of an associated measure-valued process called the exploration process, which plays a key role in most applications. Under suitable assumptions, we prove that whenever a sequence of rescaled Galton-Watson processes converges in distribution, their genealogies also converge to the continuous branching structure coded by the appropriate height process. We apply this invariance principle to various asymptotics for Galton-Watson trees. We then use the duality properties of the exploration process to compute explicitly the distribution of the reduced tree associated with Poissonnian marks in the height process, and the finite-dimensional marginals of the so-called stable continuous tree. This last calculation generalizes to the stable case a result of Aldous for the Brownian continuum random tree. Finally, we combine the genealogical structure with an independent spatial motion to develop a new approach to superprocesses with a general branching mechanism. In this setting, we derive certain explicit distributions, such as the law of the spatial reduced tree in a domain, consisting of the collection of all historical paths that hit the boundary.

Subfields: Condensed Matter Physics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8dbfa458a9cef44f0cb3bee976bac15d7b17c352

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Citations in the corpus

Limit theorems for Markov processes indexed by continuous time Galton--Watson trees (2009, background): “For all this section, we refer mainly to [18, 35, 40].”
The gap between Gromov-vague and Gromov–Hausdorff-vague topology (2014, methodology): “This approach has been successfully applied to branching forests with a particular offspring distribution (see, for example, [14,13,22]).”
Probability on Trees and Networks (2017, background): “For other aspects of turning fair games into unfair ones, see Durrett, Kesten, and Lawler (1991). 2.53. (a) 29/63, 29/35, 17/20. 2.54. (d) Let G′′ be G′ with a loop at each x of conductance π(x)pW (x, x). Then the escape probability from each x to each y is the same in G as in G′′, as is the sum of ”
Growth of Lévy trees (2005, methodology): “4 (Connection with previous the work in [10,11,26,27]) Lévy forests have first been defined in the subcritical or critical case via the coding by a process H = (Ht, t ≥ 0) introduced by Le Gall and Le Jan in [26] called the ψ-height process.”
Universal limits of substitution-closed permutation classes (2017, background): “These stable permutons are defined using the marginals of the random stable tree (see [25]), which explains the terminology.”