Heat Equation Derivative Formulas for Vector Bundles

We use martingale methods to give Bismut type derivative formulas for differentials and co-differentials of heat semigroups on forms, and more generally for sections of vector bundles. The formulas are mainly in terms of Weitzenböck curvature terms; in most cases derivatives of the curvature are not involved. In particular, our results improve B. K. Driverʹs formula in (1997, J. Math. Pures Appl. (9)76, 703–737) for logarithmic derivatives of the heat kernel measure on a Riemannian manifold. Our formulas also include the formulas of K. D. Elworthy and X.-M. Li (1998, C. R. Acad. Sci. Paris Sér. I Math.327, 87–92).