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 $\mathbb{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 $\mathbb{R} ^{+}$-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 ($\mathbb{U} $-valued) Feller diffusion and genealogical ($\mathbb{U} ^{V}$-valued) super random walk is treated based on the present work in [13] and [24].

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.