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The Markov property says that conditional on the present state, the next-state law does not depend on the earlier history. For this time-homogeneous chain, $\mathbb{P}(X_{n+1}=j\mid X_0=i_0,\dots ,X_n=i)=p_{ij}$ whenever the conditioning event has positive probability.

States communicate when each is accessible from the other by a finite positive-probability path; the equivalence classes are the communicating classes. A class is closed if no one-step transition leaves it, and open otherwise. Probability nonnegativity adds the implicit restriction $p\ge 0$ to the printed upper bound.

At $p=0$, every state is absorbing, so the classes are the singletons, all closed. For $0<p\le 1/2$, every neighbouring pair communicates, so the whole state space is one closed class, making the chain irreducible. There are no open classes in either case.

For $m=3$ and $p>0$, detailed balance along the path gives $2p{\pi }_0=p{\pi }_1$, ${\pi }_1={\pi }_2$, and $p{\pi }_2=2p{\pi }_3$. After normalization,For $0<p<1/2$, the self-loops have positive probability; the finite irreducible chain is aperiodic, so ${\mathbb{P}}_i(X_n=j)\to {\pi }_j$ for all fixed starting states and targets. For $p=1/2$, every step changes parity. The chain has period two, and for each fixed $i,j$ one parity of times has probability zero while ${\pi }_j>0$. Thus the stated convergence from a fixed starting state holds exactly when $0<p<1/2$. This is the stationary law of a reflecting nearest-neighbor walk on a finite path.