Finite detailed-balance rate and transition models

A finite rate model consists of a finite nonempty state set $S$ and rates $q_{ij}\ge 0$ in seconds${}^{-1}$ for $i\mathrm{\ne }j$. Set $q_{ii}=-{\sum }_{j\mathrm{\ne }i}{q}_{ij}$. A row probability vector is $p=(p_i{)}_{i\in S}$ with $p_i\ge 0$ and ${\sum }_i{p}_i=1$; write $\mathrm{\Delta }(S)=\{p\in [0,1{]}^S:{\sum }_i{p}_i=1\}$. A time-dependent law is a coordinatewise continuously differentiable map $p:[0,\mathrm{\infty })\to \mathrm{\Delta }(S)$, with a right derivative at $t=0$, that satisfies the continuous-time master equation ${\dot{p}}_i(t)={\sum }_{j\mathrm{\ne }i}[p_j(t)q_{ji}-p_i(t)q_{ij}]$ for every $t\ge 0$. A strictly positive normalized vector $\pi$ is a detailed-balance law when ${\pi }_i{q}_{ij}={\pi }_j{q}_{ji}$ for every distinct pair. Its undirected rate graph has an edge $\{i,j\}$ exactly when ${\pi }_i{q}_{ij}>0$; detailed balance makes this equivalent to ${\pi }_j{q}_{ji}>0$.

On a finite nonempty state set $S$, a discrete transition kernel is a matrix $K:S\times S\to [0,1]$ satisfying ${\sum }_{j\in S}K(i,j)=1$ for every $i\in S$. For a strictly positive row probability vector $\pi \in \mathrm{\Delta }(S)$, the kernel satisfies $\pi$-detailed balance when ${\pi }_{i}K(i,j)={\pi }_{j}K(j,i)$ for every $i,j\in S$. This identity pairs the supports of forward and reversed transitions. A rate model or a reversible kernel is a mathematical/physical model class; no observed trajectory is asserted by the definition.