= Uniform conditional law for a triangular-wedge walk
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 $\{0,\ldots,k\}$ 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.
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