Genealogy-valued Feller diffusion

Abstract

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, elements of the space Í. 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 develop the calculus in such a way that it can be applied in the future to more complicated systems as logistic branching or state dependent branching.We prove existence, uniqueness and Feller property of solutions of the martingale problem for this genealogy valued, i.e. Í-valued Feller diffusion. We give the precise relation to the time-inhomogeneous Í 1 -valued Fleming-Viot process. The uniqueness is shown via Feynman-Kac duality with the distance matrix augmented Kingman coalescent. Via the Feynman-Kac duality we deduce the generalized branching property of the Í-valued Feller diffusion. Using a semigroup operation on Í together with the branching property we obtain a Lévy-Khintchine formula for Í-valued Feller diffusion and determine explicitly the Lévy measure.For h > 0 we obtain a decomposition into depth-h subfamilies which leads to a representation of the in terms of a Cox point process of genealogies of single ancestor subfamilies. Furthermore, correcting a result from the Ê + -valued literature, we will identify the Í-valued process conditioned to survive until a finite time T . This is the key ingredient of the excursion law of the Í-valued Feller diffusion.We study long time asymptotics of the Í-valued Feller diffusion conditioned to survive forever, its generalized quasi-equilibrium and Kolmogorov-Yaglom limit law and show the limit processes solve well-posed Í-valued martingale problems. We also obtain various representations of the long time limits: backbone construction of the Palm distribution, the Í-valued version of the Kallenberg tree, the Í-valued version of Feller's branching diffusion with immigration from an immortal line à la Evans. This requires considering conditioned martingale problems which are of different form that those in the unconditioned case. On the level of Ívalued processes we still have equality (in law) of the Q-process, i.e., the process conditioned to survive up to time T in the limit T → ∞, the size-biased process and Evans' branching process with immigration from an immortal line. The Í-valued generalized quasi-equilibrium is a size-biased version of the Kolmogorov-Yaglom limit law.The above results are also key tools for analyzing genealogies in spatial branching populations. We construct the genealogy of the spatial version of the Feller diffusion on a countable group (super random walk). We give results on a martingale problem characterization, duality, generalized branching property and the long time behavior for this object. As an appl...