Patric Karl Glöde | Searcharxiv

arXiv2022

Longtime behavior of completely positively correlated Symbiotic Branching Model

We study the longtime behavior of a continuous state Symbiotic Branching Model (SBM). SBM can be seen as a unified model generalizing the Stepping Stone Model, Mutually Catalytic Branching Processes, and the Parabolic Anderson Model. It was introduced by Etheridge and Fleischmann in 2004. The key parameter in these models is the local correlation $ρ$ between the driving Brownian Motions. The longtime behavior of all SBM exhibits a dichotomy between coexistence and non-coexistence of the two populations depending on the recurrence and transience of the migration and also in many cases on the branching rate. The most significant gap in the understanding of the longtime behavior of SBM is for positive correlations in the transient regime. In this article we give a precise description of the longtime behavior of the SBM with $ρ=1$ with not necessarily identical initial conditions.

math.PR↗

arXiv2020

Branching Processes -- A General Concept

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.

math.PR↗