Genealogy-valued Feller diffusion (original) (raw)
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arXiv: Probability, 2019
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 mathbbU\mathbb{U}mathbbU. 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., mathbbU\mathbb{U}mathbbU-valued Feller diffusion. We give the precise relation to the time-inhomogeneous mathbbU_1\mathbb{U}_1mathbbU_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 mathbbU\mathbb{U}mathbbU, called concatenation, together with the...
Branching Processes - A General Concept
Latin American Journal of Probability and Mathematical Statistics, 2021
The paper has four goals. First, we want to generalize the classical concept of the branching property so that it becomes applicable for historical and genealogical processes (using the coding of genealogies by ($V$-marked) ultrametric measure spaces leading to state spaces mathbbU\mathbb{U}mathbbU resp. mathbbUV\mathbb{U}^VmathbbUV). The processes are defined by well-posed martingale problems. In particular we want to complement the corresponding concept of infinite divisibility developed in \cite{infdiv} for this context. Second one of the two main points, we want to find a corresponding characterization of the generators of branching processes more precisely their martingale problems which is both easy to apply and general enough to cover a wide range of state spaces. As a third goal we want to obtain the branching property of the mathbbU\mathbb{U}mathbbU-valued Feller diffusion respectively mathbbUV\mathbb{U}^VmathbbUV-valued super random walk and the historical process on countable geographic spaces the latter as two examples of...
Limit Theorems for Branching Processes with Immigration in a Random Environment
2020
We investigate subcritical Galton-Watson branching processes with immigration in a random environment. Using Goldie's implicit renewal theory we show that under general Cramer condition the stationary distribution has a power law tail. We determine the tail process of the stationary Markov chain, prove point process convergence, and convergence of the partial sums. The original motivation comes from Kesten, Kozlov and Spitzer seminal 1975 paper, which connects a random walk in a random environment model to a special Galton-Watson process with immigration in a random environment. We obtain new results even in this very special setting.