On diffusion processes with B(R², VMO) coefficients and “good” Green’s functions of the corresponding operators

The solvability in Sobolev spaces with special mixed norms is proved for nondivergence form second order parabolic equations. The usefulness of mixed norms in our problem is similar to that in the theory of Navier–Stokes equation, where mixed norms are indispensable and quite powerful. We obtain better regularity properties of solutions which allow us to apply the results to proving weak uniqueness of solutions of the corresponding stochastic Itô equations. The leading coefficients of our operators are assumed to be measurable in the time variable and two coordinates of space variables, and be almost in VMO with respect to the other coordinates. This solvability result implies the weak uniqueness of solutions of the corresponding stochastic Itô equations in the class of “good” solutions (which is nonempty). This also implies uniqueness of a Green’s function in the class of “good” ones (which is always nonempty).