RECORD
The structure of combinatorial Markov processes
Probability · Submitted 18 Mar 2016
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Together, our main theorems establish common structural features shared by all exchangeable combinatorial Feller processes, regardless of the dynamics or resident state space, thereby generalizing behaviors previously observed for exchangeable coalescent and fragmentation processes as well as other combinatorial stochastic processes.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
Every exchangeable Feller process taking values in a suitably nice combinatorial state space can be constructed by a system of iterated random Lipschitz functions. In discrete time, the construction proceeds by iterated application of independent, identically distributed functions, while in continuous time the random functions occur as the atoms of a time homogeneous Poisson point process. We further show that every exchangeable Feller process projects to a Feller process in an appropriate limit space, akin to the projection of partition-valued processes into the ranked-simplex and graph-valued processes into the space of graph limits. Together, our main theorems establish common structural features shared by all exchangeable combinatorial Feller processes, regardless of the dynamics or resident state space, thereby generalizing behaviors previously observed for exchangeable coalescent and fragmentation processes as well as other combinatorial stochastic processes. If, in addition, an exchangeable Feller process evolves on a state space satisfying the $n$-disjoint amalgamation property for all $n\geq1$, then its jump measure can be decomposed explicitly in the sense of L\'evy--It\^o--Khintchine.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Every exchangeable Feller process taking values in a suitably nice combinatorial state space can be constructed by a system of iterated random Lipschitz functions. | result 0.90 | 0.75 | 0.71 | asserted 1.00 |
| s1 | In discrete time, the construction proceeds by iterated application of independent, identically distributed functions, while in continuous time the random functions occur as the atoms of a time homogeneous Poisson point process. | method 0.81 | 0.66 | 0.09 | asserted 1.00 |
| s2 | We further show that every exchangeable Feller process projects to a Feller process in an appropriate limit space, akin to the projection of partition-valued processes into the ranked-simplex and graph-valued processes into the space of graph limits. | result 0.92 | 0.93 | 0.11 | asserted 1.00 |
| s3 | Together, our main theorems establish common structural features shared by all exchangeable combinatorial Feller processes, regardless of the dynamics or resident state space, thereby generalizing behaviors previously observed for exchangeable coalescent and fragmentation processes as well as other combinatorial stochastic processes. | interpretation 0.87 | 0.89 | 0.84 | asserted 1.00 |
| s4 | If, in addition, an exchangeable Feller process evolves on a state space satisfying the $n$-disjoint amalgamation property for all $n\geq1$, then its jump measure can be decomposed explicitly in the sense of L\'evy--It\^o--Khintchine. | result 0.96 | 0.85 | 0.09 | asserted 0.99 |
What was examined, and what was not
- arXiv listing abstract v2
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
Every exchangeable Feller process taking values in a suitably nice combinatorial state space can be constructed by a system of iterated random Lipschitz functions. In discrete time, the construction proceeds by iterated application of independent, identically distributed functions, while in continuous time the random functions occur as the atoms of a time homogeneous Poisson point process. We further show that every exchangeable Feller process projects to a Feller process in an appropriate limit space, akin to the projection of partition-valued processes into the ranked-simplex and graph-valued processes into the space of graph limits. Together, our main theorems establish common structural features shared by all exchangeable combinatorial Feller processes, regardless of the dynamics or resident state space, thereby generalizing behaviors previously observed for exchangeable coalescent and fragmentation processes as well as other combinatorial stochastic processes. If, in addition, an exchangeable Feller process evolves on a state space satisfying the $n$-disjoint amalgamation property for all $n\geq1$, then its jump measure can be decomposed explicitly in the sense of L\'evy--It\^o--Khintchine.
Human review
How the literature uses it
1 arXiv paper cite this in 1 sentence
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.