(a) Let $\left(X_{t}\right)$ be an irreducible continuous-time Markov chain on a finite or countable state space. What does it mean to say that the chain is (i) transient, (ii) recurrent, (iii) positive recurrent, (iv) null recurrent? What is the relation between equilibrium distributions and properties (iii) and (iv)?

A population of microorganisms develops in continuous time; the size of the population is a Markov chain $\left(X_{t}\right)$ with states $0,1,2, \ldots$ Suppose $X_{t}=n$. It is known that after a short time $s$, the probability that $X_{t}$ increased by one is $\lambda(n+1) s+o(s)$ and (if $n \geqslant 1$ ) the probability that the population was exterminated between times $t$ and $t+s$ and never revived by time $t+s$ is $\mu s+o(s)$. Here $\lambda$ and $\mu$ are given positive constants. All other changes in the value of $X_{t}$ have a combined probability $o(s)$.

(b) Write down the Q-matrix of Markov chain $\left(X_{t}\right)$ and determine if $\left(X_{t}\right)$ is irreducible. Show that $\left(X_{t}\right)$ is non-explosive. Determine the jump chain.

(c) Now assume that

$$\mu=\lambda$$

Determine whether the chain is transient or recurrent, and in the latter case whether it is positive or null recurrent. Answer the same questions for the jump chain. Justify your answers.