Stationary density for a Fokker-Planck equation ($L^{\ast }\rho =0$)

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.

If $q={\sigma }^{2}p$ and $q^{\prime }=2bp$, then $pLu=(q{u}_x{)}^{\prime }/2$. Insert compact cutoffs ${\chi }_R$, integrate the backward equation in time, and integrate by parts twice to move spatial derivatives onto ${\chi }_R$. The error is bounded by $t\Vert u{\Vert }_{\infty }(C{R}^{-2}\Vert q{\Vert }_1+C{R}^{-1}\Vert q^{\prime }{\Vert }_1)/2$. When these norms are finite it vanishes, proving $\int u(t)p=\int fp$ without assuming derivative behavior at infinity.