Merging for inhomogeneous finite Markov chains, part II: Nash and log-Sobolev inequalities

We study time-inhomogeneous Markov chains with finite state spaces using Nash and logarithmic-Sobolev inequalities, and the notion of $c$-stability. We develop the basic theory of such functional inequalities in the time-inhomogeneous context and provide illustrating examples.Comment: Published in at http://dx.doi.org/10.1214/10-AOP572 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Time inhomogeneous Markov chains with wave-like behavior

Starting from a given Markov kernel on a finite set $V$ and a bijection $g$ of $V$, we construct and study a time inhomogeneous Markov chain whose kernel at time $n$ is obtained from $K$ by transport of $g^{n-1}$. We show that this construction leads to interesting examples, and we obtain quantitative results for some of these examples.Comment: Published in at http://dx.doi.org/10.1214/09-AAP661 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org