ABSTRACT
The chapter presents some key analytical tools used to study the behaviors of stochastic processes on manifolds. It is concerned with the notion of a geodesic and with the construction of a distance on a manifold. The chapter provides a review of the integration theory on manifolds. It is dedicated to gradient flows and to Langevin type diffusions in Euclidian and Riemannian manifolds. The chapter is concerned with the stability properties of diffusions on manifolds in terms of the Ricci curvature using the Bochner-Lichnerowicz formulae presented earlier in and in. It is also concerned with the stability properties of diffusions on manifolds in terms of the Ricci curvature using the Bochner-Lichnerowicz formulae presented earlier.
