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The inequality $H\le I/2$ and the entropy dissipation identity for Ornstein-Uhlenbeck flow imply $H^{\prime }\le -2H$. An integrating factor proves the displayed estimate from initial entropy alone. Pinsker's inequality then gives $\Vert f_t-\gamma {\Vert }_1\le \sqrt{2H(f_0\mid \gamma )}{e}^{-t}$.

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

Use the zero entropy limit supplied in this subpart. By the fundamental theorem of calculus and the entropy dissipation identity for Ornstein-Uhlenbeck flow,Apply the relative Fisher information decay estimate starting at time $t$:Thus the requested entropy-dissipation inequality isIt is the Gaussian logarithmic Sobolev inequality along this evolution. Since $H^{\prime }\le -2H$, an integrating factor gives the entropy convergence rate for Ornstein-Uhlenbeck flow:For finite initial entropy, $f_t$ therefore converges to the stationary Gaussian density in relative entropy at rate $2$. If desired, Pinsker's inequality also converts this to the density estimate $\Vert f_t-\gamma {\Vert }_{L^1}\le \sqrt{2H(f_0\mid \gamma )}{e}^{-t}$.