Martingales, Poincaré Type Inequalities, and Deviation Inequalities

Using martingale techniques we will prove several deviation inequalities for diffusion processes in a compact Riemannian manifold and Lévy processes in euclidean space. We also deduce deviation inequalities from Poincaré type inequalities in the abstract setting of Dirichlet forms. We thus obtain, in a unified way, the theorems of Alon–Milman, Gromov–Milman, and a deviation inequality for the Laplace disribution.