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In a 1966 paper (Speed of Approach to Equilibrium for Kac's Caricature of a Maxwellian Gas, Arch. Rational Mech. Anal., Vol. 21), McKean seems to suggest that the successive derivatives of entropy $H (f) = - \int f \log f $ of a probability density $f$ along the heat flow have alternating signs. Is there anything proved today in this respect?

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  • $\begingroup$ Thank you. Are these references about f a probability density on R^n, f_t the convolution with a Gaussian kernel, and the derivatives are taken along H(f_t)? $\endgroup$
    – Dang Zheng
    Commented Feb 25, 2021 at 8:59

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For scalar random variables, the signs of the first four derivatives have been shown to be alternating. This is in "Higher order derivatives in Costa’s entropy power inequality" by Fan Cheng and Yanlin Geng. The question for higher derivatives seem open to the best of my knowledge.

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