We study a class of R^d-valued continuous strong Markov processes that are generated, only locally, by an ultra-parabolic operator with coefficients that are regular w.r.t. the intrinsic geometry induced by the operator itself and not w.r.t. the Euclidean one. The first main result is a local Ito formula for functions that are not twice-differentiable in the classical sense, but only intrinsically w.r.t. to a set of vector fields, related to the generator, satisfying the Hormander condition. The second main contribution, which builds upon the first one, is an existence and regularity result for the local transition density.