Journal of Functional Analysis 183, 42108 (2001)
        doi:10.1006jfan.2001.3746, available online at http:www.idealibrary.com on
          Heat Equation Derivative Formulas for Vector Bundles
                                              Bruce K. Driver1
              Department of Mathematics-0112, University of California at San Diego, La Jolla,
                                              California 92093-0112
                                         E-mail: drivermath.ucsd.edu
                                                       and
                                             Anton Thalmaier2
         Institut fu r Angewandte Mathematik, Universita t Bonn, Wegelerstr. 6, 53115 Bonn, Germany
                                     E-mail: antonwiener.iam.uni-bonn.de
                                           Communicated by L. Gross
                                Received April 5, 2000; accepted April 28, 2000
                Weuse martingale methods to give Bismut type derivative formulas for differen-
              tials and co-differentials of heat semigroups on forms, and more generally for sec-
              tions of vector bundles. The formulas are mainly in terms of Weitzenbo ck 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,
              703737) 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 Se r. I Math. 327, 8792).   2001 Academic Press
                Key Words: heat kernel measure; Malliavin calculus; Bismut formula; integration
              by parts; Dirac operator; de RhamHodge Laplacian; Weitzenbo ck decomposition.
                Contents.
              1. Introduction.
              2. General stochastic and geometric notation.
              3. Local martingales.
              4. The fundamental derivative formulas.
              5. Applications for compact M.
              6. Applications for non-compact M.
              7. Higher derivative formulas.
              Appendix A: Differential geometric notation and identities.
              Appendix B: Semigroup results.
              References.
           1 This research was partially supported by NSF Grants DMS 96-12651 and DMS 9971036.
           2 Research supported by Deutsche Forschungsgemeinschaft and SFB 256 (University of
        Bonn).
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        0022-123601 35.00
        Copyright  2001 by Academic Press
        All rights of reproduction in any form reserved.
                              HEAT EQUATION DERIVATIVE FORMULAS                                      43
                                         1. INTRODUCTION
           Let M be an n-dimensional oriented Riemannian manifold (not
        necessarily complete) without boundary and E a smooth Hermitian vector
        bundle over M. Denote by 1(E) the smooth sections of E. Further assume
        that L is a second order elliptic differential operator on 1(E) whose prin-
        ciple symbol is the dual of the Riemannian metric on M tensored with the
        identity section of Hom(E). In this paper we derive stochastic calculus for-
        mulas for DetL: and etLD: where :#1(E) and D is an appropriately
        chosen first order differential operator on 1(E).
           As an example of the kind of formula found in this paper, let us consider
        one representative special case. Namely suppose that M is a compact spin
        manifold, E=S is a spinor bundle over M, D is the Dirac operator on
        1(S)andL=&D2. Let scal denote the scalar curve of M. Then
                     &TD22                  &TD22
                  (e        D:)(x)=(De               :)(x)
                                         1      &(18)Tscal(X (x))dt      &1
                                     = E[e              0      t     #B T :(XT(x))],
                                        T                               T
        where Xt(x) is a Brownian motion on M starting at x#M, t is stochastic
        parallel translation along X (x)inS relative to the spin connection, B is
                                            v                                                       t
        a TxM-valued Brownian motion associated to Xt(x) and #BT is the Clifford
        multiplication of BT on Sx. This result is described in more detail in
        Section 5.2 below.
                                                                        2
                                                                 2 &TD2
           It is also possible to get a formula for D e                    : by iterating a minor
        generalization of the previous formula. For example if 0