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Marked metric measure spaces


 
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1. Title Title of document Marked metric measure spaces
 
2. Creator Author's name, affiliation, country Andrej Depperschmidt; University of Freiburg
 
2. Creator Author's name, affiliation, country Andreas Greven; University of Erlangen
 
2. Creator Author's name, affiliation, country Peter Pfaffelhuber; University of Freiburg
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Metric measure space, Gromov metric triples, Gromov- weak topology, Prohorov metric, Population model
 
3. Subject Subject classification 60B10, 05C80 (Primary) 60B05, 60B12 (Secondary)
 
4. Description Abstract 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 possible marks. We study the space of all (equivalence classes of) marked metric measure spaces for some fixed $I$. It arises as a state space in the construction of Markov processes which take values in random graphs, e.g. tree-valued dynamics describing randomly evolving genealogical structures in population models. We derive here the topological properties of the space of mmm-spaces needed to study convergence in distribution of random mmm-spaces. Extending the notion of the Gromov-weak topology introduced in (Greven, Pfaffelhuber and Winter, 2009), we define the marked Gromov-weak topology, which turns the set of mmm-spaces into a Polish space. We give a characterization of tightness for families of distributions of random mmm-spaces and identify a convergence determining algebra of functions, called polynomials.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) BMBF, DFG
 
7. Date (YYYY-MM-DD) 2011-03-27
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/1615
 
10. Identifier Digital Object Identifier 10.1214/ECP.v16-1615
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 16
 
12. Language English=en
 
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