SMF

Variations on a Theme by Bismut

Variations on a Theme by Bismut

D. W. STROOCK, O. ZEITOUNI
Variations on a Theme by Bismut
     
                
  • Année : 1996
  • Tome : 236
  • Format : Électronique
  • Langue de l'ouvrage :
    Anglais
  • Class. Math. : 35K05, 47D07, 58G11, 58G32
  • Pages : 291-301
  • DOI : 10.24033/ast.344

Let $M$ be a compact, connected, Riemannian manifold of dimension $d$, let $\{P_t:\,t>0\}$ denote the Markov semigroups on $C(M)$ determined by ${1\over 2}\Delta $, and let $p_t(x,y)$ denote the kernel (with respect to the Riemannian volume measure) for the operator $P_t$. (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 $\nabla \log (p_t(\,\cdot \,,y))$ 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.



Des problèmes avec le téléchargement?Des problèmes avec le téléchargement?
Informez-nous de tout problème que vous avez...