General criteria for the study of quasi-stationarity

For Markov processes with absorption, we provide general criteria ensuring the existence and exponential non-uniform convergence in weighted total variation norm to a quasi-stationary distribution. We also characterize subset of its domain attraction by an integrability condition, prove right eigenvector for semigroup process ergodicity Q-process. These results are applied one-dimensional multi-dimensional diffusion processes, pure jump continuous time reducible several communication classes, perturbed dynamical systems discrete evolving state spaces.