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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 the normalized standard Gaussian density,The two terms cancel, so the Ornstein-Uhlenbeck Fokker-Planck equation has stationary density for a Fokker-Planck equationEquivalently its Fokker-Planck probability current $-\nabla \gamma -v\gamma$ vanishes identically. The normalization follows from the Gaussian integral in each coordinate.

A probability density left unchanged by the Fokker-Planck equation ${\partial }_{t}f=L^{\ast }f$. Equivalently its Fokker-Planck probability current has zero divergence; the current itself need not vanish. For the Ornstein-Uhlenbeck Fokker-Planck equation, the normalized standard Gaussian density has zero current and is stationary.