We study a Markov process with two components: the first component evolves according to one of finitely many underlying Markovian dynamics, with a choice of dynamics that changes at the jump times of the second component. The second component is discrete and its jump rates may depend on the position of the whole process. Under regularity assumptions on the jump rates and Wasserstein contraction conditions for the underlying dynamics, we provide a concrete criterion for the convergence to equilibrium in terms of Wasserstein distance. The proof is based on a coupling argument and a weak form of the Harris theorem. In particular, we obtain exponential ergodicity in situations which do not verify any hypoellipticity assumption, but are not uniformly contracting either. We also obtain a bound in total variation distance under a suitable regularising assumption. Some examples are given to illustrate our result, including a class of piecewise deterministic Markov processes.

This work is concerned with the analysis of a stochastic approximation algorithm for the simulation of quasi-stationary distributions on finite state spaces. This is a generalization of a method introduced by Aldous, Flannery and Palacios. It is shown that the asymptotic behavior of the empirical occupation measure of this process is precisely related to the asymptotic behavior of some deterministic dynamical system induced by a vector field on the unit simplex. This approach provides new proof of convergence as well as precise asymptotic rates for this type of algorithm. In the last part, our convergence results are compared with those of a particle system algorithm (a discrete-time version of the Fleming-Viot algorithm).AMS 2010 Mathematical Subject Classification: 65C20, 60B12, 60J10; secondary 34F05, 60J20 Keywords: Quasi-stationary distributions -approximation method -reinforced random walks -random perturbations of dynamical systems.Outline: the next subsection introduces our main results. The proofs are in section 2. We study the dynamical system in 2.1, relate its long term behavior to the long term behavior of (x n ) n≥0

We provide quantitative estimates in total variation distance for positive semigroups, which can be non-conservative and non-homogeneous. The techniques relies on a family of conservative semigroups that describes a typical particle and Doeblin's type conditions inherited from [11] for coupling the associated process. Our aim is to provide quantitative estimates for linear partial differential equations and we develop several applications for population dynamics in varying environment. We start with the asymptotic profile for a growth diffusion model with time and space non-homogeneity. Moreover we provide general estimates for semigroups which become asymptotically homogeneous, which are applied to an age-structured population model. Finally, we obtain a speed of convergence for periodic semigroups and new bounds in the homogeneous setting. They are illustrated on the renewal equation. Contents 39References 39 2010 Mathematics Subject Classification. Primary 35B40; Secondary 47A35, 47D06, 60J80, 92D25.as t → ∞ and the ergodic behavior of the auxiliary semigroup. The proof of this Lemma is essentially an adaptation of the method in [11,12] that we extend to general semigroups in non-homogeneous environment, while they restrict their study to absorbed Markov processes. This more general semigroup setting allows us to capture a wider range of applications, like the renewal equation we consider in Section 3. Moreover, we go beyond the contraction of the auxiliary semigroup P (t) and characterize the asymptotic behavior of (M 0,t ) t≥0 , which is a novelty compared to the previous results. More precisely, for any initial time s ≥ 0, we propose conditions involving a coupling probability measure ν which guarantee the existence of a positive bounded function h s and a family of probabilities (γ t ) t≥0 such that when t → ∞ sup µ TV ≤1 µM s,t − µ(h s )ν(m s,t )γ t TV = o ν(m s,t ) .