68 research outputs found
A path-valued Markov process indexed by the ancestral mass
A family of Feller branching diffusions , , with nonlinear
drift and initial value can, with a suitable coupling over the {\em
ancestral masses} , be viewed as a path-valued process indexed by . 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 , 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 using various couplings of 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
We obtain a representation of Feller's branching diffusion with logistic
growth in terms of the local times of a reflected Brownian motion with a
drift that is affine linear in the local time accumulated by at its current
level. As in the classical Ray-Knight representation, the excursions of are
the exploration paths of the trees of descendants of the ancestors at time
, and the local time of at height measures the population size at
time (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
and living at time 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 with a sequence of Harris paths which figure
in a Ray-Knight representation of the total mass of a branching particle
system. We obtain a suitable joint convergence of together with its local
times {\em and} with the Girsanov densities that introduce the dependence in
the reproduction
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