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 $\mathbb{U}$. 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., $\mathbb{U}$-valued Feller diffusion. We give the precise relation to the time-inhomogeneous $\mathbb{U}_1$-valued Fleming-Viot process. The uniqueness is shown via Feynman-Kac duality with the distance matrix augmented Kingman coalescent. Using a semigroup operation on $\mathbb{U}$, called concatenation, together with the branching property we obtain a L{\'e}vy-Khintchine formula for $\mathbb{U}$-valued Feller diffusion and we determine explicitly the L{\'e}vy measure on $\mathbb{U}\setminus\{0\}$. From this we obtain for $h>0$ the decomposition into depth-$h$ subfamilies, a representation of the process as concatenation of a Cox point process of genealogies of single ancestor subfamilies. Furthermore, we will identify the $\mathbb{U}$-valued process conditioned to survive until a finite time $T$. 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 $\mathbb{R}$-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