In this talk we study the asymptotic behavior of the Markov chains on the projective space P(R^d) induced by an iid random walk on the general linear group GL_d(R).

In a first part, we give a qualitative description of the stationary probability measures of these Markov chains without irreducibility assumptions. The results generalize those of Bougerol-Picard (92) in the particular case of an action by affinities on R^d, and link those of Furstenberg-Kifer (82) and those of Guivarc’h-Raugi (07) and Benoist-Quint (16) in the case of an irreducible action on R^d. Based on a joint work with Cagri Sert. These results can be placed in a more general framework, that of recurrence/transience of random walks on homogeneous spaces. In the last part of the talk, we focus on the particular case of random walks by affinities on R^d and show the topological recurrence of the Markov chain in the critical case. Based on a joint work with Sara Brofferio and Marc Peigné.