Fokker–Planck stationary probability currents

Take the scalar Itô equation $dx=a(x,t)dt+b(x,t)dW_t$ and define $D=b^2/2$. Itô's formula applied to a smooth test function gives the generator $a\partial_x+D\partial_x^2$. Transferring derivatives onto the probability density gives the forward equation. The diffusion term is a second derivative of the product Dp, not D times the second derivative of p unless D is constant. Its factor one-half has already been absorbed into the definition of D. Mixing these two definitions changes predicted spreading by a factor of two.

Writing the forward equation in continuity form identifies the current as $j=ap-\partial_x(Dp)$. In one spatial dimension p has inverse-length units and j has inverse-time units: it is probability crossing a point per unit time. With variable D the current contains both a density gradient and a diffusivity-gradient term. A Stratonovich model must first be converted to its corresponding Itô drift. Using the Stratonovich drift directly in the Itô current formula generally predicts the wrong stationary state.

A stationary one-dimensional density satisfies $\partial_xj=0$, so its current is constant in space. On a reflecting interval the boundary flux is zero, forcing that constant to vanish. On a periodic ring, a nonzero constant can circulate indefinitely without accumulating probability anywhere. For constant drift v and diffusion D on a ring of circumference L, the normalized stationary density is uniform and the current is v divided by L. The density looks featureless even though trajectories retain a preferred direction of winding.

Higher-dimensional stationary systems can support divergence-free circulation with nonuniform densities. A density histogram alone therefore cannot distinguish equilibrium from a driven steady state. One must also examine currents, response, or time-reversed trajectory statistics. Boundary topology matters: a constant force around a ring is not generated by a single-valued periodic scalar potential. It represents sustained driving. Treating its uniform stationary distribution as a Boltzmann equilibrium without inspecting the circulating flux would erase the mechanism that maintains dissipation.

For zero Itô drift on a reflecting interval, zero current requires Dp to be constant. The stationary density is consequently proportional to one over D, rather than uniform. Regions of smaller diffusivity retain more probability in this particular model. This is a statement about the declared Itô process, not a universal rule for particles moving through temperature gradients or fluids with varying friction. Those physical reductions may generate additional drift terms; their microscopic assumptions must determine the correct equation.

A practical audit begins by stating the stochastic interpretation, identifying D from the noise amplitude, forming the complete current, and imposing the actual boundary condition. Substitute any proposed stationary density back into j before accepting it. Then normalize it, checking whether the required integral converges. A formal stationary expression with divergent normalization is not a probability distribution. These steps separate local algebra from physical existence and prevent the common shortcut of setting the drift to zero and assuming that the density must then be flat.