We prove a sharp rate of convergence to stationarity for a natural random walk on a compact Riemannian manifold (M, g). The proof includes a detailed study of the spectral theory of the associated operator.
"Semi-classical analysis of a random walk on a manifold." Ann. Probab. 38 (1) 277 - 315, January 2010. https://doi.org/10.1214/09-AOP483