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Random trees, Lévy processes and spatial branching processes. (English) Zbl 1037.60074
Astérisque 281. Paris: Société Mathématique de France (ISBN 2-85629-128-7/pbk). vi, 147 p. (2002).
The authors explore the connection between the genealogical structure of critical or subcritical continuous-state branching processes and a real-valued random process, called the height process. The latter is derived from the local time functional of a Lévy process with no negative jumps; its Laplace functional coincides with the branching mechanism $$\Psi$$ of the associated branching process. The case $$\Psi(u) =u^2$$ corresponding to Feller’s branching diffusion and to reflected linear Brownian motion as the associated height process has been known and studied for quite some time, and has also been exploited in the spatial context of superprocesses, where it is known as the Brownian snake construction. However, the extension to general $$\Psi$$ and the corresponding Lévy snake, is relatively recent, see J.-F. Le Gall and Y. Le Jan [Ann. Probab. 26, 213–252 (1998; Zbl 0948.60071) and ibid. 26, 1407–1432 (1998; Zbl 0945.60090)].
This book presents a rigorous and self-contained derivation of this theory and develops some new applications thereof: New properties and regularities of the height process are proven with the help of the related exploration process, a measure-valued Markov process which plays an important rule throughout this work. A convergence result – stating that the genealogies of rescaled Galton-Watson trees tend to the height process (in an appropriate sense) whenever the associated branching processes converge – is also generalized and used to render the asymptotic description of various conditioned and reduced trees. A duality property for the exploration process allows to study the stable continuous random tree (analogue to Aldous’ Brownian continuum random tree in the $$\Psi(u)=u^2$$ case), that is the reduced tree associated with Poissonian marks in the height process. Finally, new properties of the Lévy snake and its application to superprocesses, including their exit measures and reduced spatial trees, are developed.

##### MSC:
 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 34A30 Linear ordinary differential equations and systems 60G51 Processes with independent increments; Lévy processes 60J25 Continuous-time Markov processes on general state spaces 60G57 Random measures 60F17 Functional limit theorems; invariance principles 60G52 Stable stochastic processes