WIAS Preprint No. 2627, (2019)
Combinatorial considerations on the invariant measure of a stochastic matrix
Authors
- Stephan, Artur
ORCID: 0000-0001-9871-3946
2010 Mathematics Subject Classification
- 60Jxx
Keywords
- 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
DOI
Abstract
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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