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 decompositions 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 diameter 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 concatenation 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 identically 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 [DG19b] 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 forests.
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 U resp. U V ). The processes are defined by wellposed martingale problems. In particular we want to complement the corresponding concept of infinite divisibility developed in [GGR19] 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 U-valued Feller diffusion respectively U V -valued super random walk and the historical process on countable geographic spaces the latter as two examples of a whole zoo of spatial processes we could treat. The fourth goal is to show the robustness of the method and to get the branching property for genealogies marked with ancestral path, giving the line of descent moving through the ancestors and space, leading to path-marked ultrametric measure spaces. This processes are constructed here giving our second major result. The starting point for all four points is the Feller diffusion model, the final goal the (historical) super random walk model.We develop a framework covering above situations and questions, 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 objects are defined on the state space of the process. The method allows to treat every type of population model formulated as solution to a wellposed martingale problem. 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 and/or multitype models once genealogies are coded this way.We use this approach to analyze in particular the U-valued Feller diffusion, 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 main point a new model (ancestral path-)marked genealogical super random walk. This is a prototype of a situation, where historical information sits in the mark more specifically, which may contain information on the ancestral path. Another example is a multitype population, more specific with genetic types under mutation. We get here the most general model for genealogies of spatial populations in our coding. It exhibits all possibilities for obstacles we may face for the branching property which shows the robustness of the method.As method of proof for the branching property in these new examples, as the U-valued Feller diffusion or U V -valued super random walk and processes of thi...
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