Tree-valued Feller diffusion (original) (raw)

2019, arXiv: Probability

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...

Nonconservative Diffusions on [0,1] with Killing and Branching: Applications to Wright-Fisher Models with or without Selection

International Journal of Stochastic Analysis, 2011

We consider nonconservative diffusion processes on the unit interval, so with absorbing barriers. Using Doob-transformation techniques involving superharmonic functions, we modify the original process to form a new diffusion process presenting an additional killing rate part . We limit ourselves to situations for which is itself nonconservative with upper bounded killing rate. For this transformed process, we study various conditionings on events pertaining to both the killing and the absorption times. We introduce the idea of a reciprocal Doob transform: we start from the process , apply the reciprocal Doob transform ending up in a new process which is but now with an additional branching rate , which is also upper bounded. For this supercritical binary branching diffusion, there is a tradeoff between branching events giving birth to new particles and absorption at the boundaries, killing the particles. Under our assumptions, the branching diffusion process gets eventually globally...

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