Lectures on finite Markov chains. (English) Zbl 0885.60061

Saloff-Coste, Laurent

Bernard, Pierre (ed.), Lectures on probability theory and statistics. Ecole d’été de probabilités de Saint-Flour XXVI–1996. Lectures given at the Saint-Flour summer school of probability theory, August 19–September 4, 1996. Berlin: Springer. Lect. Notes Math. 1665, 301-413 (1997).

The work describes the analytic approach to the asymptotic analysis of finite continuous-time Markov chains. To obtain quantitative estimates of the convergence to the stationary distribution, the author uses functional analysis tools (such as Poincaré inequalities, Sobolev and Nash inequalities and logarithmic Sobolev inequalities) as well as geometric tools (such as geometry of graph associated with the chain and combinatorics of corresponding paths). A plenty of examples is presented. Complicated technical issues are considered in very clear and lively manner.
For the entire collection see [Zbl 0871.00040].

Reviewer: E.Kazarovitsky (Kyïv)

MSC:

60J27	Continuous-time Markov processes on discrete state spaces
60J10	Markov chains (discrete-time Markov processes on discrete state spaces)
47A25	Spectral sets of linear operators

Keywords:

Markov chain; stationary distribution; convergence; spectral gap