A continuum-tree-valued Markov process

Abstract

We present a construction of a Lévy continuum random tree (CRT) associated with a super-critical continuous state branching process using the so-called exploration process and a Girsanov's theorem. We also extend the pruning procedure to this super-critical case. Let $\psi$ be a critical branching mechanism. We set ${\psi }_{\theta }(\cdot )=\psi (\cdot +\theta )-\psi (\theta )$. Let $\Theta =({\theta }_{\infty },+\infty )$ or $\Theta =[{\theta }_{\infty },+\infty )$ be the set of values of $\theta$ for which ${\psi }_{\theta }$ is a branching mechanism. The pruning procedure allows to construct a decreasing Lévy-CRT-valued Markov process $(\ct_θ,θ\inΘ)$, such that ${\mathcal{T}}_{\theta }$ has branching mechanism ${\psi }_{\theta }$. It is sub-critical if $\theta >0$ and super-critical if $\theta <0$. We then consider the explosion time $A$ of the CRT: the smaller (negative) time $\theta$ for which ${\mathcal{T}}_{\theta }$ has finite mass. We describe the law of $A$ as well as the distribution of the CRT just after this explosion time. The CRT just after explosion can be seen as a CRT conditioned not to be extinct which is pruned with an independent intensity related to $A$. We also study the evolution of the CRT-valued process after the explosion time. This extends results from Aldous and Pitman on Galton-Watson trees. For the particular case of the quadratic branching mechanism, we show that after explosion the total mass of the CRT behaves like the inverse of a stable subordinator with index 1/2. This result is related to the size of the tagged fragment for the fragmentation of Aldous' CRT.

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• Growth of Levy trees
• Pruning Galton-Watson Trees and Tree-valued Markov Processes
• Height process for super-critical continuous state branching process
• The falling appart of the tagged fragment and the asymptotic disintegration of the Brownian height fragmentation
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Cited by in corpus (26)

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• Spaces of algebraic measure trees and triangulations of the circle
• Invariance principles for pruning processes of Galton-Watson trees
• Pruning of CRT-sub-trees
• Skeletal stochastic differential equations for superprocesses
• Local convergence of critical random trees and continuous-state branching processes
• The maximal jump and local convergence of continuous-state branching processes
• The prolific backbone for supercritical superdiffusions
• A Central Limit Theorem for Punctuated Equilibrium
• A Williams' Decomposition for Spatially Dependent Superprocesses
• Pruning a Lévy continuum random tree
• Some limit theorems for flows of branching processes
• Pruning, cut trees, and the reconstruction problem
• Totally Ordered Measured Trees and Splitting Trees with Infinite Variation II: Prolific Skeleton Decomposition
• A tree-valued Markov processes associated with an admissible family of branching mechanisms
• A construction of a $\beta$-coalescent via the pruning of Binary Trees
• Skeletal stochastic differential equations for continuous-state branching process
• Brownian continuum random tree conditioned to be large
• Combinatorial Levy processes
• Limit theorems for continuous time branching flows