Lumped Markov chain

A lumped Markov chain records only a partition class of the original state. It is Markov when the transition probability into every class is constant within each source class.

For a triangular-lattice walk in a sixty-degree wedge with rate $1/6$ on interior edges and doubled outward boundary rates, the horizontal position is conditionally uniform on $\{0,\dots ,k\}$ given its current vertical level and past levels. Within level $k$, the reflecting horizontal generator is symmetric, while vertical killing has constant rate $2/3$, so the uniform law survives any holding period. Each column of the upward or downward transition-rate block has sum $1/3$ for $k>0$, yielding a uniform conditional law at the next level. The induced birth-death process has rates ${\lambda }_k=(k+2)/(3(k+1))$ and ${\mu }_k=k/(3(k+1))$. Its probability of ever hitting zero from level $k$ is $1/(k+1)$, so it is transient.