- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:44
Revista Matemática Iberoamericana
Rev. Mat. Iberoamericana
RMI
0213-2230
2235-0616
General
10.4171/RMI
http://www.ems-ph.org/doi/10.4171/RMI
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2012)
22
2006
2
A logarithmic Sobolev form of the Li-Yau parabolic inequality
Dominique
Bakry
Université Paul Sabatier, TOULOUSE CEDEX 9, FRANCE
Michel
Ledoux
Université Paul Sabatier, TOULOUSE CEDEX 9, FRANCE
Logarithmic Sobolev inequality, Li-Yau parabolic inequality, heat semigroup, gradient estimate, non-negative curvature, diameter bound
We present a finite dimensional version of the logarithmic Sobolev inequality for heat kernel measures of non-negatively curved diffusion operators that contains and improves upon the Li-Yau parabolic inequality. This new inequality is of interest already in Euclidean space for the standard Gaussian measure. The result may also be seen as an extended version of the semigroup commutation properties under curvature conditions. It may be applied to reach optimal Euclidean logarithmic Sobolev inequalities in this setting. Exponential Laplace differential inequalities through the Herbst argument furthermore yield diameter bounds and dimensional estimates on the heat kernel volume of balls.
Global analysis, analysis on manifolds
Probability theory and stochastic processes
General
683
702
10.4171/RMI/470
http://www.ems-ph.org/doi/10.4171/RMI/470