Branching processes with competition and generalized Ray Knight Theorem

Ray-knight Theorems via Continuous Trees

In this paper, we are interested in obtaining the generalized Ray-Knight theorems (previously proved by Yor 10]) using branching processes. It was already observed in O'Connell's thesis 6] that the construction of the natural tree associated to Brownian excursion (Le Gall 4], Neveu-Pitman 5]) enables to obtain one of the "classical" Ray-Knight theorems. Our aim is to use the same ideas to get t...

Central Limit Theorem in Multitype Branching Random Walk

A discrete time multitype (p-type) branching random walk on the real line R is considered. The positions of the j-type individuals in the n-th generation form a point process. The asymptotic behavior of these point processes, when the generation size tends to infinity, is studied. The central limit theorem is proved.

Lecture 1: Generalized Ray Knight Theorem for Finite Markov Chains

From exploration paths to mass excursions – variations on a theme of Ray and Knight

Based on an intuitive approach to the Ray-Knight representation of Feller’s branching diffusion in terms of Brownian excursions we survey a few recent developments around exploration and mass excursions. One of these is Bertoin’s “tree of alleles with rare mutations” [6], seen as a tree of excursions of Feller’s branching diffusion. Another one is a model of a population with individual reprodu...

Binary trees, exploration processes, and an extented Ray–Knight Theorem

We study the bijection betwen binary Galton Watson trees in continuous time and their exploration process, both in the suband in the supercritical cases. We then take the limit over renormalized quantities, as the size of the population tends to infinity. We thus deduce Delmas’ generalization of the second Ray–Knight theorem.