The aim of this paper is to take an in-depth look at the longtime behaviour of some continuous time Markovian dynamical systems and at itsnumerical analysis. We first propose a short overview of the main ergodicity propertiesof time continuous homogeneous Markov processes (stability, positive recurrence). The basictool is a Lyapunov function. Then, we investigate if these properties still hold forthe time discretization of these processes, either with constant or decreasing step (ODE method in stochasticapproximation, Euler scheme for diffusions). We point out severaladvantages of the weighted empirical random measures associated to these procedures, especially withdecreasing step, in terms of convergence and of rate of convergence. Several simulations illustrate theseresults.
