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 …