A path-valued Markov process indexed by the ancestral mass

A family of Feller branching diffusions $Z^x$, $x \ge 0$, with nonlinear drift and initial value $x$ can, with a suitable coupling over the {\em ancestral masses} $x$, be viewed as a path-valued process indexed by $x$. 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 $Z$, 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 $x\mapsto Z^x$ using various couplings of $Z$ 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.

The tree length of an evolving coalescent

A well-established model for the genealogy of a large population in equilibrium is Kingman's coalescent. For the population together with its genealogy evolving in time, this gives rise to a time-stationary tree-valued process. We study the sum of the branch lengths, briefly denoted as tree length, and prove that the (suitably compensated) sequence of tree length processes converges, as the population size tends to infinity, to a limit process with cadlag paths, infinite infinitesimal variance, and a Gumbel distribution as its equilibrium.Comment: 23 pages, 6 figure

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

We obtain a representation of Feller's branching diffusion with logistic growth in terms of the local times of a reflected Brownian motion $H$ with a drift that is affine linear in the local time accumulated by $H$ at its current level. As in the classical Ray-Knight representation, the excursions of $H$ are the exploration paths of the trees of descendants of the ancestors at time $t=0$, and the local time of $H$ at height $t$ measures the population size at time $t$ (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 $s$ and living at time $t=H_s$ 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 $H$ with a sequence of Harris paths $H^N$ which figure in a Ray-Knight representation of the total mass of a branching particle system. We obtain a suitable joint convergence of $H^N$ together with its local times {\em and} with the Girsanov densities that introduce the dependence in the reproduction