Browsing by Author "Neumann, Pit"

A generalization of Donsker’s theorem states that, under mild conditions, Itô diffusions on complete Riemannian manifolds can be approximated by geodesic random walks. Generally, this does not provide an efficient way of simulating Itô diffusions on Riemannian manifolds, since a non-linear differential equation must be solved at every step. In the special case of Brownian motion on a compact manifold, approximate geodesic random walks based on approximations of the exponential map called retractions have been proposed and shown to be efficiently computable and convergent to Brownian motion. Using the theory of Feller processes, we generalize the method of retraction-based random walks and its convergence results to Itô diffusions on compact Riemannian manifolds under mild assumptions on the drift and diffusion terms.

We study random walks on sub-Riemannian manifolds using the framework of retractions, i.e., approximations of normal geodesics. We show that such walks converge to the correct horizontal Brownian motion if normal geodesics are approximated to at least second order. In particular, we (i) provide conditions for convergence of geodesic random walks defined with respect to normal, compatible, and partial connections and (ii) provide examples of computationally efficient retractions, e.g., for simulating anisotropic Brownian motion on Riemannian manifolds.