On nonlinear state estimation in a Riemannian manifold

We consider state estimation for partially observed processes evolving in a Riemannian manifold and obtain some new results on un-normalized nonlinear filters. We develop both local (coordinatised) versions and for the first time, extrinsic results for embedded Riemannian manifolds; the latter have computational advantages. Our results also cover continuous discrete filters. We also obtain extrinsic conditions for a process evolving in a Riemannian manifold to be Brownian motion.