Se p 20 05 Random Trees , Lévy Processes and Spatial Branching Processes

Random trees. Random trees. The main goal of this work is to investigate the genealogical structure of continuous-state branching processes in connection with limit theorems for discrete Galton-Watson trees. Applications are also given to the construction and various properties of spatial branching processes including a general class of superprocesses. Our starting point is the recent work of Le Gall and Le Jan [33] who proposed a coding of the genealogy of general continuous-state branching processes via a real-valued random process called the height process. Recall that continuous-state branching processes are the continuous analogues of discrete Galton-Watson branching processes, and that the law of any such process is characterized by a real function ψ called the branching mechanism. Roughly speaking, the height process is a continuous analogue of the contour process of a discrete branching tree, which is easy to visualize (see Section 0.1, and note that the previous informal interpretation of the height process is made mathematically precise by the results of Chapter 2). In the important special case of the Feller branching diffusion (ψ(u) = u 2), the height process is reflected linear Brow-nian motion: This unexpected connection between branching processes and Brownian motion, or random walk in a discrete setting has been known for long and exploited by a number of authors (see e. The key contribution of [33] was to observe that for a general subcritical continuous-state branching process, there is an explicit formula expressing the height process as a functional of a spectrally positive Lévy process whose Laplace exponent ψ is precisely the branching mechanism. This suggests that many problems concerning the genealogy of continuous-state branching processes can be restated and solved in terms of spectrally positive Lévy processes, for which a lot of information is available (see e.g. Bertoin's recent monograph [6]). It is the principal aim of the present work to develop such applications. In the first two sections below, we briefly describe the objects of interest in a discrete setting. In the next sections, we outline the main contributions of the present work.