Random Trees, Levy Processes and Spatial Branching Processes

Details Random Trees, Levy Processes and Spatial Branching Processes Random Trees, Levy Processes and Spatial Branching Processes

2005-09-23

arxiv.org/abs/math/0509558v1

Mathematics

Probability

Research monograph 2002; 153 pages

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Random Trees, Levy Processes and Spatial Branching Processes does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Random Trees, Levy Processes and Spatial Branching Processes, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Random Trees, Levy Processes and Spatial Branching Processes will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-71318