Fokker-Planck probability current ($J$)

Writing a Fokker-Planck equation as ${\partial }_{t}P=-{\partial }_{x}J$ defines its probability current. For ${\partial }_{t}P={\partial }_x(gP)+\frac{1}{2}{\partial }_x^2(hP)$,

Forthe probability current isso the equation is the continuity equation ${\partial }_{t}P+{\partial }_i{J}_i=0$.

When boundary terms vanish, integration by parts gives

For linear drift $A(x)=ax$ and constant symmetric positive-definite diffusion matrix $b$, a zero-current stationary Gaussian $P(x)\propto \exp (-x^{T}Dx/2)$ obeysThis reversible solution exists when $D=-2b^{-1}a$ is symmetric positive definite. Its covariance $C=D^{-1}$ satisfies the Continuous Lyapunov equation

A reflecting boundary confines a diffusion without losing probability. Its forward condition is zero outward Fokker-Planck probability current, $J\cdot n=0$, and its backward condition is a zero conormal derivative.

A zero-flux boundary condition sets the normal probability current to zero. It conserves total probability inside the domain and represents reflection for a confined diffusion.

An absorbing boundary removes a diffusion when it is reached. The forward density and backward observables satisfy Dirichlet boundary conditions there.

A partially absorbing boundary removes a particle at finite surface reactivity $\kappa$ and reflects it otherwise. Its forward Robin boundary condition is $J\cdot n=\kappa p$. If the normal diffusion coefficient is $b/2$, integration by parts gives the backward condition

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 $e^{-V}$.