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. Thr formulas are mainly in terms of Weitzenbock curvature terms; in most cases derivatives of the curvature are not involved. In particular, our results improve B. K. Driver's 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 (C) 2001 Academic Press.