For a discrete-time Markov chain, if the probability of transition $i \rightarrow j$ does not depend on $i$ then the chain is reduced to a sequence of independent random variables (states). In this case, the chain forgets about its initial state and enters equilibrium after a single transition. In the continuous-time case, a Markov chain whose rates $q_{i j}$ of transition $i \rightarrow j$ depend on $j$ but not on $i \neq j$ still 'remembers' its initial state and reaches equilibrium only in the limit as the time grows indefinitely. This question is an illustration of this property.

A protean sea sponge may change its colour among $s$ varieties $1, \ldots, s$, under the influence of the environment. The rate of transition from colour $i$ to $j$ equals $r_{j} \geqslant 0$ and does not depend on $i, i \neq j$. Consider a Q-matrix $Q=\left(q_{i j}\right)$ with entries

$$q_{i j}= \begin{cases}r_{j}, & i \neq j \\ -R+r_{i}, & i=j\end{cases}$$

where $R=r_{1}+\ldots+r_{s}$. Assume that $R>0$ and let $\left(X_{t}\right)$ be the continuous-time Markov chain with generator $Q$. Given $t \geqslant 0$, let $P(t)=\left(p_{i j}(t)\right)$ be the matrix of transition probabilities in time $t$ in chain $\left(X_{t}\right)$.

(a) State the exponential relation between the matrices $Q$ and $P(t)$.

(b) Set $\pi_{j}=r_{j} / R, j=1, \ldots, s$. Check that $\pi_{1}, \ldots, \pi_{s}$ are equilibrium probabilities for the chain $\left(X_{t}\right)$. Is this a unique equilibrium distribution? What property of the vector with entries $\pi_{j}$ relative to the matrix $Q$ is involved here?

(c) Let $\mathbf{x}$ be a vector with components $x_{1}, \ldots, x_{s}$ such that $x_{1}+\ldots+x_{s}=0$. Show that $\mathbf{x}^{\mathrm{T}} Q=-R \mathbf{x}^{\mathrm{T}}$. Compute $\mathbf{x}^{\mathrm{T}} P(t)$

(d) Now let $\delta_{i}$ denote the (column) vector whose entries are 0 except for the $i$ th one which equals 1. Observe that the $i$ th row of $P(t)$ is $\delta_{i}^{\mathrm{T}} P(t)$. Prove that $\delta_{i}^{\mathrm{T}} P(t)=\pi^{\mathrm{T}}+e^{-t R}\left(\delta_{i}^{\mathrm{T}}-\pi^{\mathrm{T}}\right) .$

(e) Deduce the expression for transition probabilities $p_{i j}(t)$ in terms of rates $r_{j}$ and their sum $R$.