Stationary Gaussian density for a linear Fokker-Planck equation

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