Abstract
Applying the stochastic calculus of variations on frame bundles along tangent processes, we derive Bismut type formulae for the derivatives of diffusion semigroups on Riemannian manifold in both variables. We also obtain the Bismut formulae expressed in terms of the Ricci and torsion tensors for the connection with torsion.
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References
Airault, H., Malliavin, P.: Integration by parts formulas and dilatation vector fields on elliptic probability spaces. Probab. Theory Related Fields 106, 447–494 (1996)
Bismut, J.M.: Large Deviations and the Malliavin Calculus. Birkhäuser, Basel (1984)
Capitaine, M., Hsu, E., Ledoux, M.: Martingale representation and a simple proof of logarithmic sobolev inequalities on path spaces. Elect. Comm. Probab. 2, 71–81 (1997)
Cruzeiro, A.B., Malliavin, P.: Renormalized differential geometry on path space: structural equation, curvature. J. Funct. Anal. 139, 119–181 (1996)
Cruzeiro, A.B., Malliavin, P.: Non perturbative construction of invariant measure through confinement by curvature. J. Math. Pures Appl. 77, 527–537 (1998)
Driver, B.K:. A Cameron–Martin type quasi-invariance theorem for Brownian motion on a compact manifold. J. Funct. Anal. 110, 272–376 (1992)
Driver, B.K.: Integration by parts for heat kernel measures revisited. J. Math. Pures Appl. 76, 703–737 (1997)
Elworthy, K.D.: Stochastic Differential Equations on Manifolds. Cambridge University Press (1982)
Elworthy, K.D., Li, X.-M.: Formulae for the derivatives of heat semigroups. J. Funct. Anal. 125, 252–286 (1994)
Elworthy, K.D., Le Jan, Y., Li, X.M.: On the Geometry of Diffusion Operators and Stochastic Flows, Lecture Notes in Math. Springer, Berlin Heidelberg New York (1720)
Fang, S.: Formule de Weitzenbock relative à une connexion à torsion, preprint
Fang, S., Malliavin, P.: Stochastic analysis on the path space of a riemannian manifold: I. Markovian Stochastic Calculus. J. Funct. Anal. 118, 249–274 (1993)
Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Process. North-Holland, Amsterdam (1981)
Malliavin, P.: Formule de la moyenne, Calcul de perturbations et théorème d'annulation pour les formes harmoniques. J. Funct. Anal. 274–291 (1974)
Norris, J.R.: Path integral formulae for heat kernels and their derivatives. Probab. Theory Related Fields 94, 525–541 (1993)
Thalmaier, A., Wang, F.-Y.: Gradient estimates for harmonic functions on regular domains in Riemannian manifolds. J. Funct. Anal. 155, 109–124 (1998)
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Cruzeiro, A.B., Zhang, X. Bismut Type Formulae for Diffusion Semigroups on Riemannian Manifolds. Potential Anal 25, 121–130 (2006). https://doi.org/10.1007/s11118-006-9010-8
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DOI: https://doi.org/10.1007/s11118-006-9010-8