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For a sufficiently regular unit-mass solution, the Ornstein-Uhlenbeck Fokker-Planck equation is ${\partial }_{t}f=\nabla \cdot (f\nabla \log (f/\gamma ))$. Integration by parts proves the displayed identity. Consequently relative entropy decreases, and its dissipation is relative Fisher information.

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

The given relative Fisher information inequality is $I^{\prime }(t)\le -2I(t)$. Multiplying by $e^{2t}$ and integrating gives relative Fisher information decay under Ornstein-Uhlenbeck flow:In particular $I(t)\le I(0)e^{-2t}$ when $I(0)$ is finite; if necessary one starts at a positive time with finite information. It follows that $I(t)\to 0$.

Write $H(t)=H(f_t\mid \gamma )$. It is nonnegative and decreasing, so has a finite limit $H_{\infty }\ge 0$ when $H(0)<\infty$. The printed integrability request concerns the product $H(t)H^{\prime }(t)$. Its sign is nonpositive, and the fundamental theorem of calculus givesPassing to $R\to \infty$ proves time integrability of an entropy-dissipation product:Also ${\int }_0^{\infty }|H^{\prime }|=H(0)-H_{\infty }\le H(0)$. The zero value of $H_{\infty }$ will be used as supplied in (f); positivity and monotonicity alone only establish existence of the limit.

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 $\Vert f_t-\gamma {\Vert }_{L^1}\le \sqrt{2H(f_0\mid \gamma )}{e}^{-t}$.