Markov Chain & Detailed Balance

Detailed Balance is the key sufficient condition for a Markov chain to have stationary distribution $\pi$: if the transition kernel $P(i \to j)$ satisfies the pairwise reversibility

then $\pi$ is a stationary distribution of the chain. Proof: for any state $j$,

the first step substitutes detailed balance into each term of the sum, the last uses row-stochasticity $\sum_i P(j \to i) = 1$ — one step from $\pi$ stays at $\pi$, i.e. $\pi^T P = \pi^T$, exactly the definition of stationarity.

the mathematical foundation of MCMC (especially Metropolis-Hastings) — designing a transition kernel reduces to verifying detailed balance pointwise, and stationarity follows automatically; interviews often ask "how does detailed balance imply a stationary distribution".

it turns "find a stationary distribution" from solving the linear system $\pi^T P = \pi^T$ (an eigenvalue problem) into checking a pairwise condition, which is far easier to construct; note that detailed balance is sufficient but not necessary — non-reversible chains can be stationary without it, yet MCMC almost always uses reversible designs (simpler and usually faster mixing).

From detailed balance $\pi(i) P(i \to j) = \pi(j) P(j \to i)$, prove $\sum_i \pi(i) P(i \to j) = \pi(j)$ — i.e. $\pi$ is stationary — justifying each step?

Is detailed balance sufficient or necessary for stationarity? Give an example of a stationary but non-reversible chain?

How does the vector form $\pi^T P = \pi^T$ follow from detailed balance?

Why do MCMC constructions use detailed balance instead of solving $\pi^T P = \pi^T$ directly?

Which regularity conditions (irreducibility, aperiodicity, positive recurrence) are additionally needed for convergence to stationarity?