Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 13C / e / Solution

{c}

For linear drift $A(x)=ax$ and constant symmetric positive-definite diffusion matrix $b$, a zero-current stationary Gaussian $P(x)\propto\exp(-x^TDx/2)$ obeys
$$
a+\frac12bD=0.
$$
This reversible solution exists when $D=-2b^{-1}a$ is symmetric positive definite. Its covariance $C=D^{-1}$ satisfies the <Continuous Lyapunov equation>
$$
aC+Ca^T+b=0.
$$

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 Incoming links: Stationary Gaussian density for a linear Fokker-Planck equation