Ornstein-Uhlenbeck Fokker-Planck equation (${\partial }_{t}f={\Delta }_{v}f+{\nabla }_v\cdot (vf)$)

The probability-density equation for the Ornstein-Uhlenbeck process with drift $-v$ and diffusion coefficient $\sqrt{2}$. The unit-covariance Gaussian density is stationary because its Fokker-Planck probability current vanishes. This normalization makes mean velocity decay at rate one and energy excess decay at rate two.

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.

Integration by parts against $1$, $v_i$ and $|v{|}^2/2$ gives these equations for mass, unnormalized momentum and kinetic energy. With $M_0\ne 0$, the mean is $u(t)=e^{-t}{u}_0$, and $E(t)=d{M}_0/2+(E_0-d{M}_0/2)e^{-2t}$. Normalizing by mass is invalid if that mass is zero.