HYPOELLIPTICITY OF THE STOCHASTIC PARTIAL-DIFFERENTIAL OPERATORS

This work is devoted to the proof of the following result using the general results of our preceding work on the nuclear space-valued semimartingales: Let u be a D'-valued semimartingale satisfying the following SPDE: du(t) = (-p + ((-1)m(i) + 1/2)q(i)2)u(t)dt + q(i)u(t)dW(t)i + dh(t) i = 1,..., N; where p and q(i) are random partial differential operators with C(infinity) coefficients, m(i) is the degree of q(i) and h is a semimartingale with values in the space of C(infinity)-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(infinity)-functions. Although the problem may seem of a particular type, the methods we use to solve it are general.