Consider fluctuations of a population described by the vector $\mathbf{x}=\left(x_{1}, x_{2}, \ldots, x_{N}\right)$. The probability of the state $\mathbf{x}$ at time $t, P(\mathbf{x}, t)$, obeys the multivariate Fokker-Planck equation

$$\frac{\partial P}{\partial t}=-\frac{\partial}{\partial x_{i}}\left(A_{i}(\mathbf{x}) P\right)+\frac{1}{2} \frac{\partial^{2}}{\partial x_{i} \partial x_{j}}\left(B_{i j}(\mathbf{x}) P\right),$$

where $P=P(\mathbf{x}, t), A_{i}$ is a drift vector and $B_{i j}$ is a symmetric positive-definite diffusion matrix, and the summation convention is used throughout.

(a) Show that the Fokker-Planck equation can be expressed as a continuity equation

$$\frac{\partial P}{\partial t}+\nabla \cdot \mathbf{J}=0$$

for some choice of probability flux $\mathbf{J}$ which you should determine explicitly. Here, $\nabla=\left(\frac{\partial}{\partial x_{1}}, \frac{\partial}{\partial x_{2}}, \ldots, \frac{\partial}{\partial x_{N}}\right)$ denotes the gradient operator.

(b) Show that the above implies that an initially normalised probability distribution remains normalised,

$$\int P(\mathbf{x}, t) d V=1$$

at all times, where the volume element $d V=d x_{1} d x_{2} \ldots d x_{N}$.

(c) Show that the first two moments of the probability distribution obey

$$\begin{aligned} \frac{d}{d t}\left\langle x_{k}\right\rangle &=\left\langle A_{k}\right\rangle \\ \frac{d}{d t}\left\langle x_{k} x_{l}\right\rangle &=\left\langle x_{l} A_{k}+x_{k} A_{l}+B_{k l}\right\rangle \end{aligned}$$

(d) Now consider small fluctuations with zero mean, and assume that it is possible to linearise the drift vector and the diffusion matrix as $A_{i}(\mathbf{x})=a_{i j} x_{j}$ and $B_{i j}(\mathbf{x})=b_{i j}$ where $a_{i j}$ has real negative eigenvalues and $b_{i j}$ is a symmetric positive-definite matrix. Express the probability flux in terms of the matrices $a_{i j}$ and $b_{i j}$ and assume that it vanishes in the stationary state.

(e) Hence show that the multivariate normal distribution,

$$P(\mathbf{x})=\frac{1}{Z} \exp \left(-\frac{1}{2} D_{i j} x_{i} x_{j}\right)$$

where $Z$ is a normalisation and $D_{i j}$ is symmetric, is a solution of the linearised FokkerPlanck equation in the stationary state, and obtain an equation that relates $D_{i j}$ to the matrices $a_{i j}$ and $b_{i j}$.

(f) Show that the inverse of the matrix $D_{i j}$ is the matrix of covariances $C_{i j}=\left\langle x_{i} x_{j}\right\rangle$ and obtain an equation relating $C_{i j}$ to the matrices $a_{i j}$ and $b_{i j}$.