Spectral analysis of hypoelliptic random walks

We study the spectral theory of a reversible Markov chain associated to a hypoelliptic random walk on a manifold M. This random walk depends on a parameter h ∈]0, h0] which is roughly the size of each step of the walk. We prove uniform bounds with respect to h on the rate of convergence to equilibrium, and the convergence when h → 0 to the associated hypoelliptic diffusion