Uniqueness of a probability solution to the Kolmogorov equation with a diffusion matrix satisfying Dini’s condition

In this note we study the stationary Kolmogorov equation and prove that, in the case where the diffusion matrix satisfies Dini’s condition and the drift coefficient is locally integrable to a power greater than the dimension, the ratio of two probability solutions belongs to the Sobolev class, and in the case of existence of a Lyapunov function or the global integrability of the coefficients with respect to the solution a probability solution is unique.