A subexponential upper bound for entropy convergence of Markov chains with a spectral gap (Q1291154)

Let \(P\) be a Markov kernel on a measurable space \(S\) with an invariant probability \(\mu\). Let \(P^*\) be the adjoint operator of \(P\) on \(L_2(\mu)\). It is assumed that the operator \(I-PP^*\) possesses a spectral gap. It is shown that this assumption implies some estimates of the speed of convergence to zero of the entropy \(\text{Ent}(P^nm\mid\mu)\), \(n\to\infty\), where \(m\) is a given probability on \(S\) satisfying certain integrability conditions, and the action \(P^nm\) is defined in a natural way. A modification of this approach is used also for some inhomogeneous Markov chains.