On diffusion processes with $B(\mathbb{R}^{2}, VMO)$ coefficients and "good" Green's functions of the corresponding operators

Abstract

The solvability in Sobolev spaces with special mixed norms is proved for nondivergence form second order parabolic equations. The leading coefficients are assumed to be measurable in the time variable and two coordinates of space variables, and be almost in VMO (vanishing mean oscillation) with respect to the other coordinates. This solvability result implies the weak uniqueness of solutions of the corresponding stochastic It\^o 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).