I have a Markov chain with two states $S= \{0,1\}$ where the transition rate $\mu,\nu>0$. The transition rate from 1 to 0 is $\nu$ and from 0 to 1 $\mu$. Initially $X_0 =0$. I want to:
- Write down the transition rate matrix $W$ of the Markov chain.
- Write down the master equation for the single-time probability distribution $p_t(n)$ for $t\geq 0$ and $n \in S$.
My Idea: The transition rate matrix is given by \begin{align*} W = \begin{pmatrix} 0 & \mu \\ \nu & 0 \end{pmatrix} \end{align*} The eigenvalues are $\lambda_{1,2} = \mp \sqrt{\mu \nu}$, which gives the eigenvectors \begin{align*} \mathbf{v}_{1}=\left[\begin{array}{c}{-\sqrt{\mu / \nu}} \\ {1}\end{array}\right], \quad \mathbf{v}_{2}=\left[\begin{array}{c}{\sqrt{\mu / \nu}} \\ {1}\end{array}\right] \end{align*} The matrixexponential is then \begin{align*} p_t(n) &= V \exp(D t) V^{-1} \\ &= \begin{bmatrix} \sqrt{\mu/\nu} & -\sqrt{\mu / \nu} \\ 1 & 1 \end{bmatrix} \begin{bmatrix} \exp(-\sqrt{\mu / \nu} t) & 0 \\ 0 & \exp(\sqrt{\mu / \nu } t) \end{bmatrix} \begin{bmatrix} -\frac{\sqrt{\nu }}{2 \sqrt{\mu }} & \frac{1}{2} \\ \frac{\sqrt{\nu }}{2 \sqrt{\mu }} & \frac{1}{2} \\ \end{bmatrix} \\ &= \mathrm{e}^{-t \sqrt{\mu \nu}} \begin{bmatrix} \frac{1}{2} \left(1+e^{2 t \sqrt{\mu \nu }}\right) & \frac{\left(-1+e^{2 t \sqrt{\mu \nu }}\right) \sqrt{\mu }}{2 \sqrt{\nu }} \\ \frac{\left(-1+e^{2 t \sqrt{\mu \nu }}\right) \sqrt{\nu }}{2 \sqrt{\mu }} & \frac{1}{2} \left(1+e^{2 t \sqrt{\mu \nu }}\right) \\ \end{bmatrix} \end{align*} Now I need the master equation, but I'm not sure how to get it. Furthermore I'm not sure if my solution is correct so far or if there is an easier way.