Branching Processes - A General Concept - Greven, Andreas; Rippl, Thomas; Glöde, Patric Karl (arXiv, 2018)
The paper has three goals. First, we want to generalize the classical concept of the branching property so that it becomes applicable in historical and genealogical processes (using the coding of genealogies by ultrametric measure spaces) and in particular complementing the corresponding concept of infinite divisibility developed in [GGR18] for this context. Second the main point, we want to find a corresponding characterization of the generators of branching processes respectively their martingale problems which is both easy to apply and general enough to cover a wide range of state spaces. The third goal is to get the branching property for genealogies marked with ancestral path, e.e. path-marked ultrametric measure spaces. The starting point for all three points is the Feller diffusion model, the final goal the super random walk model. We develop a framework covering above situations and questions and leading to a new generator criterion. The state spaces suitable here are consistent collections of topological semigroups each enriched with a collection of maps, the truncation maps all defined on the state space of the process. This framework in particular includes processes taking values in the space of marked ultrametric measure spaces and hence allows to treat historical information and genealogies of spatial population models once genealogies are coded this way. We use this approach to analyze in particular the tree-valued Feller diffusion (more precisely, equivalence class of ultrametric measure space valued), various levels of genealogies in spatial models i.e. location-marked versions of the former (as super random walk) and historical spatial branching processes or as a third main point a new model (ancestral path-)marked genealogical super random walk as most general model for genealogies of spatial populations in the mentioned coding. As method of proof for the branching property in these new examples, as the tree-valued Feller dif- fusion or tree-valued super random walk and processes of this type we work with a time-inhomogeneous martingale problem and Feynman-Kac duality to verify that the setup of the criterion is applicable.<br
Keywords: Branching property; generator criterion; genealogical processes; historical processes; processes with values in marked ultrametric measure spaces; genealogical super random walk; ances- tral path; processes with values in semigroups.

Branching trees I: Concatenation and infinite divisibility - Glöde, Patric Karl; Greven, Andreas; Rippl, Thomas (arXiv, 2017)
The goal of this work is to decompose random populations with a genealogy in subfamilies of a given degree of kinship and to obtain a notion of infinitely divisible genealogies. We model the genealogical structure of a population by (equivalence classes of) ultrametric measure spaces (um-spaces) as elements of the Polish space U which we recall. In order to then analyze the family structure in this coding we introduce an algebraic structure on um-spaces (a consistent collection of semigroups). This allows us to obtain a path of collections of subfamilies of fixed kinship h (described as ultrametric measure spaces), for every depth h as a measurable functional of the genealogy. Technically the elements of the semigroup are those um-spaces which have diam- eter less or equal to 2h called h-forests (h > 0). They arise from a given ultrametric measure space by applying maps called h−truncation. We can define a concatena- tion of two h-forests as binary operation. The corresponding semigroup is a Delphic semigroup and any h-forest has a unique prime factorization in h-trees (um-spaces of diameter less than 2h). Therefore we have a nested R + -indexed consistent (they arise successively by truncation) collection of Delphic semigroups with unique prime factorization. Random elements in the semigroup are studied, in particular infinitely divisible random variables. Here we define infinite divisibility of random genealogies as the property that the h-tops can be represented as concatenation of independent identi- cally distributed h-forests for every h and obtain a Lévy-Khintchine representation of this object and a corresponding representation via a concatenation of points of a Poisson point process of h-forests. Finally the case of discrete and marked um-spaces is treated allowing to apply the results to both the individual based and most important spatial populations. The results have various applications. In particular the case of the genealogical (U-valued) Feller diffusion and genealogical (U V -valued) super random walk is treated based on the present work in [DG18b] and [GRG]. In the part II of this paper we go in a different direction and refine the study in the case of continuum branching populations, give a refined analysis of the Laplace functional and give a representation in terms of a Cox process on h-trees, rather than forest
Keywords: Genealogy valued random variables; infinite divisibility; random trees; Cox cluster representation; Lévy-Khintchine formulas; branching processes; branching tree; (marked) ultrametric measure spaces; branching property of semigroups; random vari- ables with values in semigroups.