Destination-dependent reset chain ($Q=1r^T-RI$)

For jump rates $q_{ij}=r_j$ when $i\ne j$, assume $r_j\ge 0$ and $0<R={\sum }_j{r}_j<\infty$. The positive-rate states form the unique Closed communicating class; every zero-rate state is an open singleton. The stationary law is ${\pi }_i=r_i/R$, with Detailed balance ${\pi }_i{q}_{ij}=r_i{r}_j/R$. When at least two rates are positive, the embedded jump-chain law is proportional to $r_i(R-r_i)$. With only one positive rate, that state is absorbing and the genuine sequence of jump times stops there; an extended discrete chain can use an absorbing self-loop.