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.



Universal decay rate of the Fisher information along the heat flow
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](https://projecteuclid.org/euclid.acta/1485890415) 


