Convergence of metric two-level measure spaces

@article{Meizis2020ConvergenceOM,
title={Convergence of metric two-level measure spaces},
author={Roland Meizis},
journal={Stochastic Processes and their Applications},
year={2020}
}
• Roland Meizis
• Published 2 May 2018
• Mathematics
• Stochastic Processes and their Applications
Abstract We extend the notion of metric measure spaces to so-called metric two-level measure spaces (m2m spaces): An m2m space ( X , r , ν ) is a Polish metric space ( X , r ) equipped with a two-level measure ν ∈ M f ( M f ( X ) ) , i.e. a finite measure on the set of finite measures on X . We introduce a topology on the set of (equivalence classes of) m2m spaces induced by certain test functions (i. e. the initial topology with respect to these test functions) and show that this topology is…
5 Citations

Figures from this paper

Metric two-level measure spaces: a state space for modeling evolving genealogies in host-parasite systems
We extend the notion of metric measure spaces to so-called metric two-level measure spaces (m2m spaces for short). An m2m space is an isomorphism class of a triple (X, r, ν), where (X, r) is a
Tree-valued Feller diffusion
• Mathematics
• 2019
We consider the evolution of the genealogy of the population currently alive in a Feller branching diffusion model. In contrast to the approach via labeled trees in the continuum random tree world,
Long-Time Behavior of a PDE Replicator Equation for Multilevel Selection in Group-Structured Populations
• Biology, Mathematics
• 2021
This work considers a hyperbolic PDE model of a group-structured population, in which members within a single group compete with each other for individual-level replication; while the group also competes against other groups for group- level replication, and derives a threshold level of the relative strength of between-group competition.
Two level branching model for virus population under cell division.
• Mathematics
• 2020
In this paper we study a two-level branching model for virus populations under cell division. We assume that the cells are carrying virus populations which evolve as a branching particle system with
Inference with selection, varying population size and evolving population structure: Application of ABC to a forward-backward coalescent process with interactions
• Computer Science, Biology
• 2019
A stochastic birth-death model with competitive interaction to describe an asexual population is presented, an inferential procedure for ecological, demographic and genetic parameters are developed and an Approximate Bayesian Computation framework is developed to use the model for analyzing genetic data.

References

SHOWING 1-10 OF 58 REFERENCES
Marked metric measure spaces
• Mathematics
• 2011
A marked metric measure space (mmm-space) is a triple $(X,r,μ)$, where $(X,r)$ is a complete and separable metric space and $μ$ is a probability measure on $X \times I$ for some Polish space $I$ of
Convergence in distribution of random metric measure spaces (Λ-coalescent measure trees)
• Mathematics
• 2009
We consider the space of complete and separable metric spaces which are equipped with a probability measure. A notion of convergence is given based on the philosophy that a sequence of metric measure
Ricci curvature for metric-measure spaces via optimal transport
• Mathematics
• 2004
We dene a notion of a measured length space X having nonnegative N-Ricci curvature, for N 2 [1;1), or having1-Ricci curvature bounded below byK, forK2 R. The denitions are in terms of the
Convergence of Random Processes and Limit Theorems in Probability Theory
The convergence of stochastic processes is defined in terms of the so-called “weak convergence” (w. c.) of probability measures in appropriate functional spaces (c. s. m. s.).Chapter 1. Let $\Re$ be
Pathwise construction of tree-valued Fleming-Viot processes
In a random complete and separable metric space that we call the lookdown space, we encode the genealogical distances between all individuals ever alive in a lookdown model with simultaneous multiple
Tree-valued resampling dynamics Martingale problems and applications
• Mathematics
• 2008
The measure-valued Fleming–Viot process is a diffusion which models the evolution of allele frequencies in a multi-type population. In the neutral setting the Kingman coalescent is known to generate
Tree-valued Fleming–Viot dynamics with mutation and selection
• Mathematics
• 2012
The Fleming-Viot measure-valued diffusion is a Markov process describing the evolution of (allelic) types under mutation, selection and random reproduction. We enrich this process by genealogical
On the geometry of metric measure spaces. II
AbstractWe introduce a curvature-dimension condition CD (K, N) for metric measure spaces. It is more restrictive than the curvature bound $\underline{{{\text{Curv}}}} {\left( {M,{\text{d}},m} A Course in Metric Geometry Preface This book is not a research monograph or a reference book (although research interests of the authors influenced it a lot)—this is a textbook. Its structure is similar to that of a graduate On the geometry of metric measure spaces AbstractWe introduce and analyze lower (Ricci) curvature bounds$ \underline{{Curv}} {\left( {M,d,m} \right)} $⩾ K for metric measure spaces$ {\left( {M,d,m} \right)} \$. Our definition is based on