Summary
The heat kernel and its derivatives of a vector Laplacian on the sections of a bundle over a compact Riemannian manifold are expressed as products of the scalar heat kernel of the manifold and path integrals over the Brownian bridge. The small-time asymptotics of these integrals are computed.
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Norris, J.R. Path integral formulae for heat kernels and their derivatives. Probab. Th. Rel. Fields 94, 525–541 (1993). https://doi.org/10.1007/BF01192562
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DOI: https://doi.org/10.1007/BF01192562