Branching processes and Fleming-Viot processes are two main models in stochastic population theory. Incorporating an immigration in both models, we generalize the results of Shiga (1990) and Birkner et al. (2005) which respectively connect the Feller diffusion with the classical Fleming-Viot process and the α-stable continuous state branching process with the Beta(2 − α, α)-generalized Fleming-Viot process. In a recent work, a new class of probabilitymeasure valued processes, called M -generalized Fleming-Viot processes with immigration, has been set up in duality with the so-called M -coalescents. The purpose of this article is to investigate the links between this new class of processes and the continuous-state branching processes with immigration. In the specific case of the α-stable branching process conditioned to be never extinct, we get that its genealogy is given, up to a random time change, by a Beta(2 − α, α − 1)coalescent.
We define a Markov process in a forward population model with backward
genealogy given by the $\Lambda$-coalescent. This Markov process, called the
fixation line, is related to the block counting process through its hitting
times. Two applications are discussed. The probability that the $n$-coalescent
is deeper than the $(n-1)$-coalescent is studied. The distribution of the
number of blocks in the last coalescence of the $n$-$\operatorname
{Beta}(2-\alpha,\alpha)$-coalescent is proved to converge as
$n\rightarrow\infty$, and the generating function of the limiting random
variable is computed.Comment: Published at http://dx.doi.org/10.1214/14-AAP1077 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
43 pagesInternational audienceWe present a genealogy for superprocesses with a non-homogeneous quadratic branching mechanism, relying on a weighted version of the superprocess and a Girsanov theorem. We then decompose this genealogy with respect to the last individual alive (William's decomposition). Letting the extinction time tend to infinity, we get the Q-process by looking at the superprocess from the root, and define another process by looking from the top. Examples including the multitype Feller diff usion and the superdiffusion are provided
We perform various changes of measure in the lookdown particle system of Donnelly and Kurtz. The first example is a product type h-transform related to conditioning a Generalized Fleming Viot process without mutation on coexistence of some genetic types in remote time. We give a pathwise construction of this h-transform by just "forgetting" some reproduction events in the lookdown particle system. We also provide an intertwining relationship for the Wright Fisher diffusion and explicit the associated pathwise decomposition. The second example, called the linear or additive h-transform, concerns a wider class of measure valued processes with spatial motion. Applications include: -a simple description of the additive h-transform of the Generalized Fleming Viot process, which confirms a suggestion of Overbeck for the usual Fleming Viot process -an immortal particle representation for the additive h-transform of the Dawson Watanabe process.
We establish a phase transition for the parking process on critical Galton-Watson trees. In this model, a random number of cars with mean m and variance σ 2 arrive independently on the vertices of a critical Galton-Watson tree with finite variance Σ 2 conditioned to be large. The cars go down the tree towards the root and try to park on empty vertices as soon as possible. We show a phase transition depending onSpecifically, when m ≤ 1, if Θ > 0, then all but (possibly) a few cars will manage to park, whereas if Θ < 0, then a positive fraction of the cars will not find a spot and exit the tree through the root. This confirms a conjecture of Goldschmidt and Przykucki [8].
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