Heat Equation Derivative Formulas for Vector Bundles

By B. Driver, Anton Thalmaier. 2001

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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 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 (1998, C. R. Acad. Sci. Paris Ser. I Math.327, 87–92).

Subfields: Applied Mathematics, Mathematical Physics

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