I'm looking for a reference for the following fact: In the torus $\mathbb T^d$ let me denote by $u_t=u(t,x)$ the (unique, distributional) solution of the heat equation $$ \partial_t u=\Delta u $$ started from an arbitrary probability distribution $u_0\in\mathcal P(\mathbb T^d)$. I know that there is a universal constant (probably depending on the dimension only?) such that the Fisher information $$ \mathcal F(u)=\int_{\mathbb T^d}\nabla\log u^2 u $$ decays at a linear rate, $$ \mathcal F(u_t)\leq \frac{C}{t},\qquad \forall \,t>0. $$ The point is that $C$ does not depend on $u_0$ (as long as it is normalized to be a probability measure). Unfortunately I cannot seem to find a precise reference, so any help would be greatly appreciated.
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2$\begingroup$ This can be seen by integrating the LiYau inequality: Theorem 1.1 in projecteuclid.org/euclid.acta/1485890415 $\endgroup$ – Fabrice Baudoin Sep 23 '18 at 20:48

1$\begingroup$ Super, merci Fabrice ;) Please make that comment an aswer, so I can mark it as accepted? $\endgroup$ – leo monsaingeon Sep 23 '18 at 21:16

$\begingroup$ I put the comment as an answer. $\endgroup$ – Fabrice Baudoin Sep 24 '18 at 2:23
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The result actually holds with $C=d/2$ on any compact Riemannian manifold with a nonnegative Ricci curvature. This can be seen by integrating the LiYau inequality: Theorem 1.1 in