 Reflected lazy walk with doubled endpoint transitions

On the finite path $\{0,\dots ,m\}$, take interior neighbor probabilities $p$, endpoint inward probability $2p$, and holding probability $1-2p$, with $0<p\le 1/2$. Detailed balance gives the displayed invariant distribution: endpoint weights are half the interior weights. For $p<1/2$ the chain is irreducible and aperiodic, so transition probabilities converge to it. At $p=1/2$ parity alternates and the period is two.