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A logarithmic Sobolev form of the Li-Yau parabolic inequality. (English) Zbl 1116.58024

In the very interesting paper under review the authors present a finite dimensional version of the logarithmic Sobolev inequality for heat kernel measures of non-negatively curved diffusion operators which contains and improves the Li-Yau parabolic inequality from [P. Li and S. T. Yau, Acta Math. 156, 154–201 (1986; Zbl 0611.58045)]. That new inequality is of interest already in Euclidean space for the standard Gaussian measure and the result may also be seen as an extended version of the semigroup commutator properties under curvature conditions.

MSC:

58J35 Heat and other parabolic equation methods for PDEs on manifolds
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60J45 Probabilistic potential theory

Citations:

Zbl 0611.58045
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References:

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