Branching trees I: concatenation and infinite divisibility

Abstract

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.

Branching Processes - A General Concept

“…In other words, can we define an abstract generalized branching property dealing with this problem. This complements the investigation of a concept of infinite divisibility for genealogical structures modeled as ultrametric measure spaces which is introduced in Glöde et al (2019) which presents a generalized infinite divisibility since the classical one does not hold.…”

“…Example 2.2 (Ultrametric measure space U-valued processes). Recall the setting of Glöde et al (2019) and Depperschmidt and Greven (2019) using equivalence classes of ultrametric measure spaces and the semigroup of t-forests. In that paper the evolving genealogy of the population alive at time t was described via a set of individuals U t , the genealogical distance between individuals r t (•, •) on U t × U t , a population size µ t and a sampling (probability) measure µ t on U t , altogether giving an ultrametric measure space (U t , r t ,μ t µ t ) and finally with its isomorphy class [U t , r t ,μ t µ t ] we get the elements of the state space U describing for us genealogies.…”

“…The branching property is intimitely connected with the concept of infinite divisibility. In Glöde et al (2019) we developed a new concept of infinite divisibility for genealogy valued processes, which used some abstract algebraic and fitting topological structures which allowed to prove basic facts on infinitely divisible random variables with states in genealogies. This allowed to prove the (generalized form of) Lévy -Khintchine representation of genealogy valued random variables.…”

“…U V ). In particular we want to complement the corresponding concept of infinite divisibility developed in Glöde et al (2019) for this context. The processes we consider are always defined by well-posed martingale problems.…”

“…We focus here on the description of genealogical information more in the spirit of the description via historical processes (Dawson and Perkins, 1991) or R-trees (Evans et al, 2006), by using here equivalence classes of (marked) ultrametric measure spaces. The latter approach is developed in Greven et al (2009), Depperschmidt et al (2011), applied to Fleming-Viot type models in Depperschmidt et al (2012) and Greven et al (2013) and it has been extended to branching in Glöde (2012); Depperschmidt and Greven (2019); Glöde et al (2019). This particular description of genealogies is chosen in this paper to be able to work with the framework of well-posed martingale problems to characterize the models and second to be able to use the concept of infinite divisibility from Glöde et al (2019).…”