Branching Processes -- A General Concept

TL;DR

This paper generalizes the concept of branching processes to complex genealogical and spatial models, providing new characterizations, demonstrating the branching property in various examples, and establishing the robustness of the approach.

Contribution

It introduces a broad framework for genealogical and spatial branching processes using ultrametric measure spaces, with new characterizations and proofs of the branching property.

Findings

Generalized branching property for genealogical processes

Characterization of generators for broad state spaces

Established branching property for spatial and historical processes

Abstract

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 $\mathbb{U}$ resp. $\mathbb{U}^V$). 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 $\mathbb{U}$-valued Feller diffusion respectively $\mathbb{U}^V$-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 ultra-metric 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.