= Stationary law of a reflecting nearest-neighbor walk on a finite path
On states $0,\ldots,m$, take an interior step of $+1$ or $-1$ with probability $p$ each and hold with probability $1-2p$. At an endpoint take the inward step with probability $2p$ and hold with probability $1-2p$. For $m\geq1$ and $0<p\leq1/2$, the chain is <irreducible Markov chain>[irreducible] and its <stationary distribution> has endpoint weights $1/(2m)$ and interior weights $1/m$.
<Detailed balance> proves this: the doubled inward endpoint transition requires its stationary mass to be half that of an adjacent interior point, and all interior points have equal mass. The total unnormalized weight is $1+2(m-1)+1=2m$. For $m=1$ there are just two equal endpoint masses.
When $p<1/2$, positive self-loops make the chain <aperiodic Markov chain>[aperiodic], so its transition probabilities converge to this law. At $p=1/2$ every move changes parity and two-step returns are possible, giving period two and no such convergence from a fixed starting state. At $p=0$ every state is absorbing and uniqueness of the stationary law is lost.
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