Universal decay rate of the Fisher information along the heat flow

Question

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.

Answer 1

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

On the parabolic kernel of the Schrödinger operator