68 research outputs found

    A path-valued Markov process indexed by the ancestral mass

    Full text link
    A family of Feller branching diffusions ZxZ^x, x≥0x \ge 0, with nonlinear drift and initial value xx can, with a suitable coupling over the {\em ancestral masses} xx, be viewed as a path-valued process indexed by xx. For a coupling due to Dawson and Li, which in case of a linear drift describes the corresponding Feller branching diffusion, and in our case makes the path-valued process Markovian, we find an SDE solved by ZZ, which is driven by a random point measure on excursion space. In this way we are able to identify the infinitesimal generator of the path-valued process. We also establish path properties of x↦Zxx\mapsto Z^x using various couplings of ZZ with classical Feller branching diffusions.Comment: 23 pages, 1 figure. This version will appear in ALEA. Compared to v1, it contains amendmends mainly in Sec. 2 and in the proof of Proposition 4.

    Trees under attack: a Ray-Knight representation of Feller's branching diffusion with logistic growth

    Full text link
    We obtain a representation of Feller's branching diffusion with logistic growth in terms of the local times of a reflected Brownian motion HH with a drift that is affine linear in the local time accumulated by HH at its current level. As in the classical Ray-Knight representation, the excursions of HH are the exploration paths of the trees of descendants of the ancestors at time t=0t=0, and the local time of HH at height tt measures the population size at time tt (see e.g. \cite{LG4}). We cope with the dependence in the reproduction by introducing a pecking order of individuals: an individual explored at time ss and living at time t=Hst=H_s is prone to be killed by any of its contemporaneans that have been explored so far. The proof of our main result relies on approximating HH with a sequence of Harris paths HNH^N which figure in a Ray-Knight representation of the total mass of a branching particle system. We obtain a suitable joint convergence of HNH^N together with its local times {\em and} with the Girsanov densities that introduce the dependence in the reproduction
    • …