Lyapunov criteria for uniform convergence of conditional distributions of absorbed Markov processes

We study the uniform convergence to quasi-stationarity of multidimensional processes absorbed when one coordinates vanishes. Our results cover competitive or weakly cooperative Lotka–Volterra birth and death Feller diffusions with interaction. To this aim, we develop an original non-linear Lyapunov criterion involving two functions, which applies general Markov processes.