Hypoellipticity of the stochastic partial differential operators

Abstract

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 q_i are random partial differential operators with C ^∞ coefficientsm_i is the degree of q_i 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