# Existence of mark functions in marked metric measure spaces

@article{Kliem2015ExistenceOM,
title={Existence of mark functions in marked metric measure spaces},
author={Sandra Kliem and Wolfgang Lohr},
journal={Electronic Journal of Probability},
year={2015},
volume={20},
pages={1-24}
}
• Published 5 December 2014
• Mathematics
• Electronic Journal of Probability
We give criteria on the existence of a so-called mark function in the context of marked metric measure spaces (mmm-spaces). If an mmm-space admits a mark function, we call it functionally-marked metric measure space (fmm-space). This is not a closed property in the usual marked Gromov-weak topology, and thus we put particular emphasis on the question under which conditions it carries over to a limit. We obtain criteria for deterministic mmm-spaces as well as random mmm-spaces and mmm-space…
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
Invariance principles for random walks in random environment on trees
In arXiv:1609.05666v1 [math.PR] a functional limit theorem was proved. It states that symmetric processes associated with resistance metric measure spaces converge when the underlying spaces converge
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,
Scaling limits of the three-dimensional uniform spanning tree and associated random walk
• Mathematics
The Annals of Probability
• 2021
We show that the law of the three-dimensional uniform spanning tree (UST) is tight under rescaling in a space whose elements are measured, rooted real trees, continuously embedded into Euclidean
The Aldous chain on cladograms in the diffusion limit
• Mathematics
• 2020
In [Ald00], Aldous investigates a symmetric Markov chain on cladograms and gives bounds on its mixing and relaxation times. The latter bound was sharpened in [Sch02]. In the present paper we encode
Continuum Space Limit of the Genealogies of Interacting Fleming-Viot Processes on $\Z$
• Mathematics
• 2015
We study the evolution of genealogies of a population of individuals, whose type frequencies result in an interacting Fleming-Viot process on $\Z$. We construct and analyze the genealogical structure
A representation for exchangeable coalescent trees and generalized tree-valued Fleming-Viot processes
We give a de Finetti type representation for exchangeable random coalescent trees (formally described as semi-ultrametrics) in terms of sampling iid sequences from marked metric measure spaces. We
A macroscopic view of two discrete random models
• Mathematics
• 2020
This thesis investigates the large-scale behaviour emerging in two discrete models: the uniform spanning tree on Z3 and the chase-escape with death process. Uniform spanning trees We consider the
Invariance principles for tree-valued Cannings chains
We consider sequences of tree-valued Markov chains that describe evolving genealogies in Cannings models, and we show their convergence in distribution to tree-valued Fleming-Viot processes. Under
Convergence of tree-valued Cannings chains
We consider sequences of tree-valued Markov chains that describe evolving genealogies in Cannings models, and we show their convergence in distribution to tree-valued Fleming-Viot processes. Under

## References

SHOWING 1-10 OF 30 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
Equivalence of Gromov-Prohorov- and Gromov's Box-Metric on the Space of Metric Measure Spaces
The space of metric measure spaces (complete separable metric spaces with a probability measure) is becoming more and more important as state space for stochastic processes. Of particular interest is
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
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
Path-properties of the tree-valued Fleming-Viot process
• Mathematics
• 2012
We consider the tree-valued Fleming–Viot process, (Xt )t≥0 , with mutation and selection. This process models the stochastic evolution of the genealogies and (allelic) types under resampling,
Probability Theory I
These notes cover the basic definitions of discrete probability theory, and then present some results including Bayes' rule, inclusion-exclusion formula, Chebyshev's inequality, and the weak law of
A compact containment result for nonlinear historical superprocess approximations for population models with trait-dependence
We consider an approximating sequence of interacting population models with branching, mutation and competition. Each individual is characterized by its trait and the traits of its ancestors. Birth-
Coalescents with multiple collisions
k−2 � 1 − xb−k � � dx� . Call this process a � -coalescent. Discrete measure-valued processes derived from the � -coalescent model a system of masses undergoing coalescent collisions. Kingman's