Bismut Type Formulae for Diffusion Semigroups on Riemannian Manifolds

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.