Define what is meant by a communicating class and a closed class in a Markov chain.

A Markov chain $\left(X_{n}: n \geqslant 0\right)$ with state space $\{1,2,3,4\}$ has transition matrix

$$P=\left(\begin{array}{cccc} \frac{1}{2} & 0 & \frac{1}{2} & 0 \\ 0 & \frac{1}{2} & 0 & \frac{1}{2} \\ \frac{1}{2} & 0 & \frac{1}{2} & 0 \\ \frac{1}{2} & 0 & 0 & \frac{1}{2} \end{array}\right)$$

Write down the communicating classes for this Markov chain and state whether or not each class is closed.

If $X_{0}=2$, let $N$ be the smallest $n$ such that $X_{n} \neq 2$. Find $\mathbb{P}(N=n)$ for $n=1,2, \ldots$ and $\mathbb{E}(N)$. Describe the evolution of the chain if $X_{0}=2$.