Romain Abraham

Pruning Galton–Watson trees and tree-valued Markov processes

We present a new pruning procedure on discrete trees by adding marks on the nodes of trees. This procedure allows us to construct and study a tree-valued Markov process

Changing the branching mechanism of a continuous state branching process using immigration

\{ {\cal G}(u)\}

Sur la mesure de sortie du super mouvement brownien

by pruning Galton-Watson trees and an analogous process

Representations of the brownian snake with drift

\{{\cal G}^*(u)\}

Critical Multi-type Galton–Watson Trees Conditioned to be Large

by pruning a critical or subcritical Galton-Watson tree conditioned to be infinite. Under a mild condition on offspring distributions, we show that the process

An Active Curve Approach for Tomographic Reconstruction of Binary Radially Symmetric Objects

\{{\cal G}(u)\}

Asymptotics for the small fragments of the fragmentation at nodes

run until its ascension time has a representation in terms of

Reflecting Brownian snake and a Neumann-Dirichlet problem

\{{\cal G}^*(u)\}

On the connected components of the support of super Brownian motion and of its exit measure

. A similar result was obtained by Aldous and Pitman (1998) in the special case of Poisson offspring distributions where they considered uniform pruning of Galton-Watson trees by adding marks on the edges of trees.

3D Image Segmentation and Cylinder Recognition for Composite Materials

We construct a continuous state branching process with immigration (CBI) whose immigration depends on the CBI itself and we recover a continuous state branching process (CB). This provides a dual construction of the pruning at nodes of CB introduced by the authors in a previous paper. This construction is a natural way to model neutral mutation. Using exponential formula, we compute the probability of extinction of the original type population in a critical or sub-critical quadratic branching, conditionally on the non extinction of the total population.