In this talk, we construct a random walk on a closed Riemannian manifold associated with a second-order subelliptic differential operator and prove its convergence to equilibrium. The construction relies on a local reduction to an operator with constant coefficients, using a technique of Fefferman and Phong based on Calderón–Zygmund localization. Convergence to equilibrium is then obtained through the spectral theory of the associated Markov operator.

Quantum mechanics is well approximated by classical physics when Planck's constant is considered small, i.e., in the semi-classical limit. Typically, one can study an observable associated with a particle, such as its momentum or its position, and show that its dynamics is given by classical dynamics at first order, with corrections of the order of Planck's constant. In this talk, I will present more precisely the concept of semi-classical limits, the standard mathematical results known for non-relativistic quantum mechanics, and my work that concerns the semi-classical limit in the context of relativistic quantum mechanics. Concretely, I will show how to adapt the modulated energy method to the Klein-Gordon and Klein-Gordon-Maxwell equations and how to recover relativistic mechanics (instead of classical mechanics) at the semi-classical limit.