On conservation of probability and the Feller property

Abstract

It is known that any smooth, nondegenerate, second-order elliptic operator on a manifold (dimension $\not= 2$) has the form $\Delta +B$, where B is a vector field and $\Delta$ is the Laplace-Beltrami operator under some Riemannian metric on the manifold. In this paper we give several conditions on the "Ricci curvature" Ric $-\nabla_B^s$ associated with the operator $\Delta + B$ to ensure that the diffusion semigroup generated by $\Delta + B$ conserves probability and possesses the Feller property.