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On estimation of the logarithmic Sobolev constant and gradient estimates of heat semigroups
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  • Published: May 1997

On estimation of the logarithmic Sobolev constant and gradient estimates of heat semigroups

  • Feng-Yu Wang1 

Probability Theory and Related Fields volume 108, pages 87–101 (1997)Cite this article

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Summary.

This paper presents some explicit lower bound estimates of logarithmic Sobolev constant for diffusion processes on a compact Riemannian manifold with negative Ricci curvature. Let Ric≧−K for some K>0 and d, D be respectively the dimension and the diameter of the manifold. If the boundary of the manifold is either empty or convex, then the logarithmic Sobolev constant for Brownian motion is not less than

max {(d d+2) d 1 2(d+1)D 2 exp [−1−(3d+2)D 2 K],     (d−1 d+1) d K exp [−4D√d K]} .

Next, the gradient estimates of heat semigroups (including the Neumann heat semigroup and the Dirichlet one) are studied by using coupling method together with a derivative formula modified from [11]. The resulting estimates recover or improve those given in [7, 21] for harmonic functions.

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Authors and Affiliations

  1. Department of Mathematics, Beijing Normal University, Beijing 100875, P.R. China (wangfy@maths.warwick.ac.uk), , , , , , CN

    Feng-Yu Wang

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  1. Feng-Yu Wang
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Received: 19 September 1995 / In revised form 11 April 1996

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Wang, FY. On estimation of the logarithmic Sobolev constant and gradient estimates of heat semigroups. Probab Theory Relat Fields 108, 87–101 (1997). https://doi.org/10.1007/s004400050102

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  • Issue Date: May 1997

  • DOI: https://doi.org/10.1007/s004400050102

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  • Mathematics Subject Classification (1991): 35S15
  • 60J60
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