Entropy dissipation along the Ornstein–Uhlenbeck semigroup: $\frac{d}{dt}\int P_tf\log P_tf\,d\gamma=-\int ((P_tf)')^2/P_tf\,d\gamma$

Let $\gamma=N(0,1)$ be the standard Gaussian measure on $\mathbb{R}$. For $s\in\mathbb{R}$ and a function $h$, let

be the Ornstein–Uhlenbeck (Mehler) semigroup. Let $\delta>0$ and $C\in\mathbb{R}$, and let $f\in C^1(\mathbb{R})$ satisfy $\delta\le f\le C$ and $|f'|\le C$. Then for every $t>0$ the function $s\mapsto\int P_sf\log P_sf\,d\gamma$ is differentiable at $t$, and

where $(P_tf)'$ is the derivative of $x\mapsto P_tf(x)$.

This is the de Bruijn-type identity behind the Bakry–Émery proof of the Gaussian log-Sobolev inequality: the entropy decreases with rate equal to the Fisher information. The proof has three ingredients:

• differentiation under the integral sign;
• the generator identity $\partial_tP_tf=(P_tf)''-x(P_tf)'$, valid for $t>0$ even though only $f\in C^1$ is assumed, because Gaussian integration by parts in the noise variable gives $(P_tf)''=\frac{e^{-2t}}{\sqrt{1-e^{-2t}}}\int y\,f'(e^{-t}x+\sqrt{1-e^{-2t}}\,y)\,d\gamma(y)$;
• Gaussian integration by parts in $x$.

Formalization Note. $P_sf$ is written out explicitly. The restriction $t>0$ is needed, since $\sqrt{1-e^{-2s}}$ is not differentiable at $s=0$.