
Anglais
Let M be a compact, connected, Riemannian manifold of dimension d, let {Pt:t>0} denote the Markov semigroups on C(M) determined by 12Δ, 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 ∇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.
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