Consider a bivariate diffusion process with drift vector $A_{i}(\mathbf{x})=a_{i j} x_{j}$ and diffusion matrix $b_{i j}$ where

$$a_{i j}=\left(\begin{array}{cc} -1 & 1 \\ -2 & -1 \end{array}\right), \quad b_{i j}=\left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right)$$

$\mathbf{x}=\left(x_{1}, x_{2}\right)$ and $i, j=1,2$.

(i) Write down the Fokker-Planck equation for the probability $P\left(x_{1}, x_{2}, t\right)$.

(ii) Plot the drift vector as a vector field around the origin in the region $\left|x_{1}\right|<1$, $\left|x_{2}\right|<1$.

(iii) Obtain the stationary covariances $C_{i j}=\left\langle x_{i} x_{j}\right\rangle$ in terms of the matrices $a_{i j}$ and $b_{i j}$ and hence compute their explicit values.