Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 1 / 26I / i / Solution

A continuous-time Markov chain started with distribution $\pi$ is reversible if its finite-dimensional distributions are unchanged by reversing time: for every $T>0$, $(X_t:0\le t\le T)$ and $(X_{T-t}:0\le t\le T)$ have the same law. In particular, reversibility includes stationarity. The detailed balance equations areFor a finite continuous-time Markov chain, these equations are equivalent to reversibility with initial distribution $\pi$. Equivalently, they imply ${\pi }_i{P}_{ij}(t)={\pi }_j{P}_{ji}(t)$ for every transition matrix $P(t)=e^{tQ}$.