Markov property and cokernels of local operators

We discuss Gaussian generalized random fields indexed by smooth sec­tions of vector bundles with respect to Markov properties. We propose a new set-up which is suitable for the present question and within which new phenomena are de­tected naturally. In particular, we give a counterexample to the belief that locality in the RKHS implies germ Markov property. We also prove the close connection between Markov property and cokernels of local operators. Furthermore we prove the Markov property for a very degenerate Gaussian random field