Multivariate Fokker-Planck probability current ($J_i$)

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