Filtering on manifolds

We coasider a partially observable diffusion process (x_t,y_t)t0 whose unobservable componentx_t lives on a submanifold M ofRⁿ. We present some general conditions under which the conditional law ofx_t, given the observationsy_s,s [0,t], admits a density w.r.t. a given measure on M. We characterize the analytical properties of this density by using appropriate Sobolev spaces.Research supported by the Hungarian National Foundation of Scientific Research No. 2290.