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.
"Merging for inhomogeneous finite Markov chains, part II: Nash and log-Sobolev inequalities." Ann. Probab. 39 (3) 1161 - 1203, May 2011. https://doi.org/10.1214/10-AOP572