While looking at one paper about Metropolic Hasting optimal convergence rates, I came accross a discrete time generator of Markov chain. It is defined as follows: $$G V(x)=nE\left [ \left( V(y)-V(x)\right )\left(1\wedge \frac{\pi(y)}{\pi(x)}\right ) \right ],$$
where $n$ is the dimension of $x$, $y$ is the proposed value by proposal distribution, expectation is taken w.r.t. the proposal distribution and $\pi$ is the target distribution.
My question is where could I find more information about the construction of discrete time generators? I know about the continuous time generators for diffusion processes, but cannot find information about the discrete analogue. A paper or (better) a textbook reference would be highly appreciated.
