 Isothermal diffusion with position-dependent drag

With local thermal noise matched to spatially varying drag, the overdamped probability density obeys the displayed diffusion equation. Its equivalent Itô diffusion is $dX=D^{\prime }(X)dt+\sqrt{2D(X)}dW$, since the two terms in the Fokker-Planck equation combine to ${\partial }_x(D{P}_x)$. Thus $d⟨X⟩/dt=⟨D^{\prime }(X)⟩$, even with no applied force. The drift points toward greater diffusivity, but a uniform density has zero flux: the drift is balanced by the spatial variation of the noise. Holding the force-noise amplitude fixed instead would not produce this isothermal Fickian equation.