# Universal decay rate of the Fisher information along the heat flow

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.

• This can be seen by integrating the Li-Yau inequality: Theorem 1.1 in projecteuclid.org/euclid.acta/1485890415 – Fabrice Baudoin Sep 23 '18 at 20:48
• Super, merci Fabrice ;-) Please make that comment an aswer, so I can mark it as accepted? – leo monsaingeon Sep 23 '18 at 21:16
• I put the comment as an answer. – Fabrice Baudoin Sep 24 '18 at 2:23

The result actually holds with $$C=d/2$$ on any compact Riemannian manifold with a non-negative Ricci curvature. This can be seen by integrating the Li-Yau inequality: Theorem 1.1 in