Variations on a Theme by Bismut

Let M be a compact, connected, Riemannian manifold of dimension d, let fPt : t > 0g denote the Markov semigroups on C (M) determined by 1 2 , and let pt (x; y) denote the kernel (with respect to the Riemannian volume measure) for the operator Pt. (The existence of this kernel as a positive, smooth function is well-known, see e.g. D].) Bismut's celebrated formula, presented in B], equates r log ? pt (; y) with certain stochastic integrals (see (20) below.) Various derivations of this formula and its extensions can be found in AM], EL] and N]. In this note, we give a quick derivation of Bismut's and related formulae by lifting considerations to the bundle of orthonormal frames, using Bochner's identity, and applying a little elementary stochastic analysis. Some consequences of these identities are then explored. In particular, after deriving a standard logarithmic Sobolev inequality, we present (see (26)) a sharp pointwise estimate on the logarithmic derivative of the heat kernel in terms of known estimates on the heat kernel itself. x1 Bismut's Formula and Variations Let O(M) denote the bundle of orthonormal frames associated to M, equipped with the L evi{ Civita connection. (Throughout, we will take our basic reference for diierential geometry to be the book BC]. In particular, see Chapter 7 for an explanation of O(M).) The advantage gained by moving considerations to O(M) is that many diierential geometric quantities resemble their classical analogs.