 Solution

For positive smooth $f$ with the stated decay, integration by parts givesand, using unit mass,For zero values of $f$, these calculations can be made with the positive unit-mass approximation $(f+\epsilon \gamma )/(1+\epsilon )$ and then passed to the limit whenever the quantities are finite. Positive-time solutions also have the usual Gaussian smoothing.

Since $-\log \gamma =v^2/2+\frac{1}{2}\log (2\pi )$, mass conservation givesThe derivative of the first term is $\int {\partial }_t{f}_t\log f_t$, because $\int {\partial }_t{f}_t=0$. The two identities above and the energy equation in (b), with $d=M=1$, yieldNow expand the relative Fisher information:where $\int v{f}^{\prime }=-1$. Thus the entropy dissipation identity for Ornstein-Uhlenbeck flow is