Branching Processes - A General Concept

Beitrag in einer Fachzeitschrift
(Online-Publikation)


Details zur Publikation

Autorinnen und Autoren: Greven A, Rippl T, Glöde PK
Zeitschrift: arXiv
Jahr der Veröffentlichung: 2018
ISSN: 2331-8442
Sprache: Englisch


Abstract

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.


FAU-Autorinnen und Autoren / FAU-Herausgeberinnen und Herausgeber

Glöde, Patric Karl
Department Mathematik
Greven, Andreas Prof. Dr.
Lehrstuhl für Mathematische Stochastik
Rippl, Thomas Dr.
Lehrstuhl für Mathematische Stochastik


Zitierweisen

APA:
Greven, A., Rippl, T., & Glöde, P.K. (2018). Branching Processes - A General Concept. arXiv.

MLA:
Greven, Andreas, Thomas Rippl, and Patric Karl Glöde. "Branching Processes - A General Concept." arXiv (2018).

BibTeX: 

Zuletzt aktualisiert 2019-24-07 um 07:24