  We prove that under H\"ormander's type conditions on the coefficients of the
unobservable component of a partially observable diffusion process the
filtering density is infinitely differentiable and can be represented as the
integral of an infinitely differentiable kernel against the prior initial
distribution. These results are derived from more general results obtained for
SPDEs. One the main novelty of the paper is the existence and smoothness of the
kernel, another that we allow the coefficients of our partially observable
process to be just measurable with respect to the time variable.
