Subexponential lower bounds for f-ergodic Markov processes

We provide a criterion for establishing lower bound on the rate of convergence in f-variations of an ergodic Markov processes to its invariant measure. The criterion consists of super and submartingale conditions for certain functionals of the Markov process. Our results complement the existing literature on the stability of Markov processes, based on Lyapunov drift conditions, which primarily focuses on upper bounds. We apply our theory to elliptic diffusions, L\'evy-driven Ornstein-Uhlenbeck processes and hypoelliptic stochastic Hamiltonian models with known polynomial/stretched exponential upper bounds on their rates of convergence. Our lower bounds match asymptotically the known upper bounds for these classes of models, establishing their rate of convergence to stationarity.