F. [. Benaïm, B. Bouguet, . Cloez-]-m, B. Benaïm, and . Cloez, Ergodicity of inhomogeneous Markov chains through asymptotic pseudotrajectories ArXiv e-prints A stochastic approximation approach to quasi-stationary distributions on finite spaces, Electron. Commun. Probab, vol.23, issue.14 6, pp.24-39, 2015.

]. M. Ben97 and . Benaïm, Vertex-reinforced random walks and a conjecture of Pemantle, Ann. Probab, vol.25, issue.1, pp.361-392, 1997.

]. M. Ben99 and . Benaïm, Dynamics of stochastic approximation algorithms Asymptotic pseudotrajectories and chain recurrent flows, with applications, Séminaire de Probabilités, XXXIII, volume 1709 of Lecture Notes in Math, pp.1-68, 1996.

T. [. Bakhtin and . Hurth, Invariant densities for dynamical systems with random switching, Nonlinearity, vol.25, issue.10, pp.2937-2952, 2012.
DOI : 10.1088/0951-7715/25/10/2937

M. Benaïm, S. Le-borgne, F. Malrieu, and P. Zitt, Quantitative ergodicity for some switched dynamical systems, Electronic Communications in Probability, vol.17, issue.0, pp.14-24, 2012.
DOI : 10.1214/ECP.v17-1932

M. Benaïm, S. Le-borgne, F. Malrieu, and P. Zitt, Qualitative properties of certain piecewise deterministic Markov processes, Annales de l'Institut Henri Poincar??, Probabilit??s et Statistiques, vol.51, issue.3, pp.1040-1075, 2015.
DOI : 10.1214/14-AIHP619

. Bmp-+-15-]-f, F. Bouguet, F. Malrieu, C. Panloup, J. Poquet et al., Long time behavior of Markov processes and beyond Exponential ergodicity for Markov processes with random switching, ESAIM Proc. Surv. Bernoulli, vol.51, issue.211 9, pp.193-211505, 2015.

T. [. Costantini and . Kurtz, DIFFUSION APPROXIMATION FOR TRANSPORT PROCESSES WITH GENERAL REFLECTION BOUNDARY CONDITIONS, Mathematical Models and Methods in Applied Sciences, vol.10, issue.05, pp.717-762, 1993.
DOI : 10.1103/RevModPhys.71.313

]. R. Dob53 and . Dobru?in, Limit theorems for a Markov chain of two states. Izvestiya Akad, Nauk SSSR. Ser. Mat, vol.17, issue.2, pp.291-330, 1953.

S. [. Dietz, . N. Sethuraman-]-s, T. G. Ethier, and . Kurtz, Occupation laws for some time-nonhomogeneous Markov chains, Algorithmes stochastiques of Mathématiques & Applications (Berlin) [Mathematics & Applications Markov processes. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical StatisticsFGM12] J. Fontbona, H. Guérin, and F. Malrieu. Quantitative estimates for the long-time behavior of an ergodic variant of the telegraph process. Adv. in Appl. Probab, pp.661-683, 1986.
DOI : 10.1214/EJP.v12-413
URL : http://arxiv.org/abs/math/0701798

]. G. For15 and . Fort, Central limit theorems for stochastic approximation with controlled Markov chain dynamics, ESAIM Probab. Stat, vol.19, pp.60-80, 2015.

]. F. Gan59, . Gantmachergou97-]-r, and . Gouet, The Theory of Matrices Strong convergence of proportions in a multicolor Pólya urn, J. Appl. Probab, vol.2, issue.342 3, pp.426-435, 1959.

S. [. Ikeda and . Watanabe, Stochastic differential equations and diffusion processes, 1989.

]. G. Jon04 and . Jones, On the Markov chain central limit theorem, Probab. Surv, vol.1, issue.3, pp.299-320, 2004.

]. V. Kol11 and . Kolokoltsov, Markov processes, semigroups and generators, De Gruyter Studies in Mathematics, vol.38, issue.9, 2011.

H. Kunita, Stochastic differential equations and stochastic flows of diffeomorphisms, École d'été de probabilités de Saint-Flour, XII?1982 Stochastic approximation and recursive algorithms and applications, pp.143-303, 1984.
DOI : 10.1007/BF00535284

P. [. Métivier and . Priouret, Th??or??mes de convergence presque sure pour une classe d'algorithmes stochastiques ?? pas d??croissant, Probability Theory and Related Fields, vol.I, issue.41, pp.403-428, 1987.
DOI : 10.1007/BF00699098

]. [. Saloff-costesv05, S. R. Sethuraman, and . Varadhan, Lectures on finite Markov chains, Lectures on probability theory and statistics (Saint-Flour, pp.301-413, 1996.
DOI : 10.1103/PhysRevLett.58.86