Entropy dissipation identity for Ornstein-Uhlenbeck flow ($\frac{d}{dt}H(f_t\mid \gamma )=-I(f_t\mid \gamma )$)

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.

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 $\Vert f_t-\gamma {\Vert }_1\le \sqrt{2H(f_0\mid \gamma )}{e}^{-t}$.

If finite nonnegative entropy $H(t)$ decreases, its limit $H_{\infty }$ exists. The fundamental theorem of calculus applied to $H^2$ proves the displayed absolute-integrability identity. This product conclusion does not require a zero limiting entropy.