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PreprintCC BY 4.0arXiv

The Markov Accessibility Topology and its Probabilistic Structure

  • Lourival Lima
  • Jorge Vielma
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Abstract

In this paper, we associate a canonical topology with the transition operator of a discrete-time Markov chain. We prove that this construction defines an Alexandroff topology, called the Markov Accessibility Topology, and subsequently establish that it coincides with the Alexandroff topology induced by the accessibility relation. This establishes a direct connection between probabilistic accessibility and order-theoretic topology, allowing several classical notions from Markov chain theory to be interpreted in topological terms. Among other results, we characterize absorbing subsets as closed sets, relate irreducibility to topological connectedness and the triviality of the topology, describe minimal open neighborhoods through accessibility and hitting probabilities, and investigate the communication quotient together with its relationship to lumpability.

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Cite

BibTeXRIS
BibTeX
@misc{lima2026markov,
  title = {{The Markov Accessibility Topology and its Probabilistic Structure}},
  author = {Lourival Lima and Jorge Vielma},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.10591},
  eprint = {2610.10591},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.10591},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The Markov Accessibility Topology and its Probabilistic Structure
AU  - Lourival Lima
AU  - Jorge Vielma
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.10591
UR  - https://arxiv.org/abs/2610.10591
AB  - In this paper, we associate a canonical topology with the transition operator of a discrete-time Markov chain. We prove that this construction defines an Alexandroff topology, called the Markov Accessibility Topology, and subsequently establish that it coincides with the Alexandroff topology induced by the accessibility relation. This establishes a direct connection between probabilistic accessibility and order-theoretic topology, allowing several classical notions from Markov chain theory to be interpreted in topological terms. Among other results, we characterize absorbing subsets as closed sets, relate irreducibility to topological connectedness and the triviality of the topology, describe minimal open neighborhoods through accessibility and hitting probabilities, and investigate the communication quotient together with its relationship to lumpability.
ER  -