Uniqueness of solutions of elliptic equations and uniqueness of invariant measures of diffusions

Let $M$ be a complete connected Riemannian manifold of dimension $d$ and let $L$ be a second order elliptic operator on $M$ that has a representation $L=a^{ij}\partial_{x_i}\partial_{x_j}+b^i\partial_{x_i}$ in local coordinates, where $a^{ij}\in H^{p,1}_{\mathrm{loc}}$, $b^i\in L^p_{\text{loc}}$ for some $p>d$, and the matrix $(a^{ij})$ is non-singular. The aim of the paper is the study of the uniqueness of a solution of the elliptic equation $L^*\mu=0$ for probability measures $\mu$, which is understood in the weak sense: $\displaystyle\int L\varphi f\,d\mu=0$ for all $\varphi\in C_0^\infty(M)$. In addition, the uniqueness of invariant probability measures for the corresponding semigroups $(T_t^\mu)_{t\geqslant 0}$ generated by the operator $L$ is investigated. It is proved that if a probability measure $\mu$ on $M$ satisfies the equation $L^*\mu=0$ and $(L-I)\bigl(C^\infty_0(M)\bigr)$ is dense in $L^1(M,\mu)$, then $\mu$ is a unique solution of this equation in the class of probability measures. Examples are presented (even with $a^{ij}=\delta^{ij}$ and smooth $b^i$) in which the equation $L^*\mu=0$ has more than one solution in the class of probability measures. Finally, it is shown that if $p>d+2$, then the semigroup $(T_t)_{t\geqslant 0}$ generated by $L$ has at most one invariant probability measure.