Potential condition for a Fokker--Planck equation

For current $J=aP-\kappa \nabla P/2$, a scalar potential exists when the one-form ${\kappa }^{-1}a$ is exact. Then $D=\kappa /2$, $a=-D\nabla V$, and $J=-PD\nabla (V+\log P)$.

For positive-definite $D$ and no-flux boundaries,decreases as $\dot{\mathcal{F}}=-\int P\nabla (V+\log P{)}^{T}D\nabla (V+\log P)dx$. Its unique normalized minimizer is proportional to $e^{-V}$.