Dirichlet forms and applications to ergodic theory of Markov chains

Using Malliavin calculus and Dirichlet forms theory we study the absolute continuity of Markov chains ergodic measures. Both discrete and continuous case are studied. Establishing a strong reinforcement of energy image density, we are able to provide with speed of convergence to equilibrium of the distributions of the chain. Various consequences are deduced of this property such as the Rajchman property of the distributions of non degenerated random vector of Dirichlet spaces.