a1 Applied Mathematics Institute, Shanghai Institute of Railway Technology, 1 Zhennan Road, Shanghai 200333, People's Republic of China
In the first part of this paper, Yau's estimates for positive L-harmonic functions and Li and Yau's gradient estimates for the positive solutions of a general parabolic heat equation on a complete Riemannian manifold are obtained by the use of Bakry and Emery's theory. In the second part we establish a heat kernel bound for a second-order differential operator which has a bounded and measurable drift, using Girsanov's formula.
(Received March 26 1993)
(Revised January 24 1994)
* This research work was supported in part by a Royal Society Fellowship and the National Natural Science Foundation of China.