Asymptotic and spectral properties of exponentially -ergodic Markov processes

For L p convergence rates of a time homogeneous Markov process, sufficient conditions are given in terms of an exponential ϕ -coupling. This provides sufficient conditions for L p convergence rates and related spectral and functional properties (spectral gap and Poincaré inequality) in terms of appropriate combination of ‘local mixing’ and ‘recurrence’ conditions on the initial process, typical in the ergodic theory of Markov processes. The range of applications of the approach includes processes that are not time-reversible. In particular, sufficient conditions for the spectral gap property for the Lévy driven Ornstein–Uhlenbeck process are established.