Hypoellipticity of the stochastic partial differential operators

This work is devoted to the proof of the following result using the general results of our preceeding work on the nuclear space-valued semimartingales: Let u be a D ′-valued semimartingale satisfying the following SPDE: ; where p and qi are random partial differential operators with C ∞ coefficientsmi is the degree of qi and h is a semimartingale with values in the space of C ∞-functions. If p satisfies a Garding's inequality on the space of the semimartingales with values in D and indexed with the uniformly bounded random intervals, then u is a semimartingale with values in the space of C ∞-functions. Although the problem may seem of a particular type, the methods we use to solve it are general