Convergence of Markov processes

The aim of this minicourse is to provide a number of tools that allow one to de-termine at which speed (if at all) the law of a diffusion process, or indeed a rather general Markov process, approaches its stationary distribution. Of particular in-terest will be cases where this speed is subexponential. After an introduction to the general ergodic theory of Markov processes, the first part of the course is devoted to Lyapunov function techniques. The second part is then devoted to an elementary introduction to Malliavin calculus and to a proof of Hörmander’s famous ”sums of squares ” regularity theorem. 1 General (ergodic) theory of Markov processes In this note, we are interested in the long-time behaviour of Markov processes, both in discrete and continuous time. Recall that a discrete-time Markov process x on a state space X is described by a transition kernel P, which we define as a measurable map from X into the space of probability measures on X. In all that follows, X will always be assumed to be a Polish space, that is a complete, separable metric space. When viewed as a measurable space, we will always endow it with its natural Borel σ-algebra, that is the smallest σ-algebra containing all open sets. This ensures that X endowed with any probability measure is a Lebesgue space and that basic intuitive results of probability and measure theory (Fubini’s theorem, regular conditional probabilities, etc) are readily available. We will sometimes consider P as a linear operator on the space of signed mea-sures on X and / or the space of bounded measurable functions by (Pµ)(A)