A Generalization of Khasminskii's Theorem on the Existence of Invariant Measures for Locally Integrable Drifts
Vladimir Igorevich Bogachev, Michael Röckner · Theory of Probability and Its Applications · 2001
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, $A^{ij}\in H^{p,1}_{\rm loc}(\mathbf{R}^n)$, $B^i\in L^p_{\rm loc}(\mathbf{R}^n)$, 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 $H^{p,1}_{\rm loc}(\mathbf{R}^n)$ 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.