Coupling,convergence rates of Markov processes and weak Poincaré inequalities

Some analytic and probabilistic properties of the weak Poincare inequality are obtained. In particular, for strong Feller Markov processes the existence of this inequality is equivalent to each of the following: (i) the Liouville property (or the irreducibility); (ii) the existence of successful couplings (or shift-couplings); (iii) the convergence of the Markov process in total variation norm; (iv) the triviality of the tail (or the invariant) σ-field; (v) the convergence of the density. Estimates of the convergence rate in total variation norm of Markov processes are obtained using the weak Poincare inequality.