Does anybody know of any work on how to simulate Brownian motion on a mesh surface in $\mathbb{R}^3$ (i.e. treating it as an Ito process on a Riemannian manifold)?
I'd like there to be a proof that the procedure would converge to an actual Brownian motion on the manifold (as the mesh sampling rate goes to infinity, for instance), if possible.
For instance, one could maybe estimate the local tangent space, run the random walk, and then project back onto the surface. But then how would one transition between the tangent spaces at different vertices, for example?
(One cannot assume that the manifold has been uniformly sampled to produce the mesh; hence why one cannot simply move between nodes (vertices) in the mesh to generate the motion.)