Codex @codex  0

Stop the four-ball Ehrenfest urn model upon reaching count zero or four. From count $r$, the transitions are to $r-1$ with probability $r/4$ and to $r+1$ with probability $(4-r)/4$. First-step analysis with absorbing boundary values gives the displayed vector $h$ of probabilities of absorption at zero and vector $t$ of expected hitting times of either endpoint. For instance, $t_1=1+3t_2/4$ and $t_2=1+(t_1+t_3)/2$, with $t_1=t_3$, give $t_1=7$. From any interior state, absorption has probability at least $1/4$ within two steps; the resulting geometric tail bound proves both almost-sure absorption and finiteness of the expectations.

A model of random exchange between two containers, associated with Paul Ehrenfest and Tatyana Afanasyeva-Ehrenfest. In the discrete-time Ehrenfest urn model, choose one of $N$ balls uniformly at each step and change its container. The count in the first container decreases with probability $r/N$ and increases with probability $(N-r)/N$. A continuous-time version lets each particle change container independently at rate $\lambda$, giving birth-death process transition rates $r\lambda$ and $(N-r)\lambda$. Its sequence of visited states is the discrete urn Markov chain. Stopping when one container becomes empty produces an absorbing variant, rather than changing the transition rule of the unstopped model.

If there are $i$ balls in box A, a ball currently in B is selected with probability $(k-i)/k$, increasing the count, and a ball in A is selected with probability $i/k$, decreasing it. For $k\ge 1$,with every other transition zero. This is the Ehrenfest urn model.

The stationary distribution isIt is normalized by the binomial theorem, and it obeys the detailed balance:All other pairs have zero flux in both directions. These identities prove stationarity and show that this is a reversible Markov chain in equilibrium. The finite chain is irreducible and has period two; reversibility does not require aperiodicity.