WIAS Preprint No. 2627, (2019)

Combinatorial considerations on the invariant measure of a stochastic matrix


  • Stephan, Artur
    ORCID: 0000-0001-9871-3946

2010 Mathematics Subject Classification

  • 60Jxx


  • Markov chain, Markov process, invariant measure, stationary measure, stationary distribution, Theorem of Frobenius-Perron, Kirchhoff tree theorem, Markov tree theorem, directed and undirected acyclic graphs, spanning trees, detailed balance




The invariant measure is a fundamental object in the theory of Markov processes. In finite dimensions a Markov process is defined by transition rates of the corresponding stochastic matrix. The Markov tree theorem provides an explicit representation of the invariant measure of a stochastic matrix. In this note, we given a simple and purely combinatorial proof of the Markov tree theorem. In the symmetric case of detailed balance, the statement and the proof simplifies even more.

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