A generalization of Khasminskii's theorem on the existence of invariant measures for locally integrable drifts

Let $A=(A^{ij})$ be a mapping with values in the space of the nonnegative symmetric operators on $\mathbf{R}^n$ and let $B=(B^i)$ be a Borel vector field on $\mathbf{R}^n$ such that $A$ is locally uniformly nondegenerate, $A^{ij}\in H^{p,1}_{\mathrm{loc}}(\mathbf{R}^n)$, $B^i\in L^p_{\mathrm{loc}}(\mathbf{R}^n)$, where $p>n$. We show that the existence of a Lyapunov function for the operator $L_{A,B}f=\sum A^{ij}\partial_{x_i}\partial_{x_j} f +\sum B^i\partial_{x_i}f$ is sufficient for the existence of a probability measure $\mu$ with a strictly positive continuous density in the class $H^{p,1}_{\mathrm{loc}}(\mathbf{R}^n)$ such that $\mu$ satisfies $L_{A,B}^{*}\mu =0$ in the weak sense and is an invariant measure for the diffusion with the generator $L_{A,B}$ on domain $C_0^\infty (\mathbf{R}^n)$. For arbitrary continuous nondegenerate $A$ and locally bounded $B$, we prove the existence of absolutely continuous solutions. An analogous generalization of Khasminskii's theorem is obtained for manifolds.