Combinatorial considerations on the invariant measure of a stochastic matrix
- Stephan, Artur
2010 Mathematics Subject Classification
- 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.