109 research outputs found
Tree-valued Feller diffusion
We consider the evolution of the genealogy of the population currently alive
in a Feller branching diffusion model. In contrast to the approach via labeled
trees in the continuum random tree world, the genealogies are modeled as
equivalence classes of ultrametric measure spaces, the elements of the space
. This space is Polish and has a rich semigroup structure for the
genealogy. We focus on the evolution of the genealogy in time and the large
time asymptotics conditioned both on survival up to present time and on
survival forever. We prove existence, uniqueness and Feller property of
solutions of the martingale problem for this genealogy valued, i.e.,
-valued Feller diffusion. We give the precise relation to the
time-inhomogeneous -valued Fleming-Viot process. The uniqueness
is shown via Feynman-Kac duality with the distance matrix augmented Kingman
coalescent. Using a semigroup operation on , called concatenation,
together with the branching property we obtain a L{\'e}vy-Khintchine formula
for -valued Feller diffusion and we determine explicitly the
L{\'e}vy measure on . From this we obtain for
the decomposition into depth- subfamilies, a representation of the process
as concatenation of a Cox point process of genealogies of single ancestor
subfamilies. Furthermore, we will identify the -valued process
conditioned to survive until a finite time . We study long time asymptotics,
such as generalized quasi-equilibrium and Kolmogorov-Yaglom limit law on the
level of ultrametric measure spaces. We also obtain various representations of
the long time limits.Comment: 93 pages, replaced by revised versio
Genealogy of catalytic branching models
We consider catalytic branching populations. They consist of a catalyst
population evolving according to a critical binary branching process in
continuous time with a constant branching rate and a reactant population with a
branching rate proportional to the number of catalyst individuals alive. The
reactant forms a process in random medium. We describe asymptotically the
genealogy of catalytic branching populations coded as the induced forest of
-trees using the many individuals--rapid branching continuum limit.
The limiting continuum genealogical forests are then studied in detail from
both the quenched and annealed points of view. The result is obtained by
constructing a contour process and analyzing the appropriately rescaled version
and its limit. The genealogy of the limiting forest is described by a point
process. We compare geometric properties and statistics of the reactant limit
forest with those of the "classical" forest.Comment: Published in at http://dx.doi.org/10.1214/08-AAP574 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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