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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). 42 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
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