A diffusion approach to Stein's method on Riemannian manifolds

We detail an approach to develop Stein's method for bounding integral metrics on probability measures defined on a Riemannian manifold mathbf M. Our approach exploits the relationship between the generator of a diffusion on mathbf M with target invariant measure and its characterising Stein operator. We consider a pair of such diffusions with different starting points, and through analysis of the distance process between the pair, derive Stein factors, which bound the solution to the Stein equation and its derivatives. The Stein factors contain curvature-dependent terms and reduce to those currently available for mathbb Rm, and moreover imply that the bounds for mathbb Rm remain valid when mathbf M is a flat manifold