Marches al\'eatoires et \'el\'ements contractants sur des espaces CAT(0)

Abstract

This thesis is dedicated to random walks on spaces with non-positive curvature. In particular, we study the case of group actions on CAT(0) spaces that admit contracting elements, that is, whose properties mimic those of loxodromic isometries in Gromov-hyperbolic spaces. In this context, we prove several limit laws, among which the almost sure convergence to the boundary without moment assumption, positivity of the drift and a central limit theorem. In a second part, we study boundary maps and stationary measures on affine buildings of type $\tilde{A}_2$, and we show that there always exists a hyperbolic isometry for a non-elementary action by isometries on such a space. Our approach involves the use of hyperbolic models for CAT(0) spaces, which were constructed by H.~Petyt, D.~Spriano and A.~Zalloum, and measured boundary theory, whose principles come from H.~Furstenberg.