I have a continuous-state discrete-time Markov chain and I want to find the density of an invariant measure when I already know by some theorems that there exists a stationary measure.
In a discrete-state context, I know how to proceed, I have some equations to solve with the matrix. I also know in the context of continuous-time and continuous-state how to proceed: In general I have the SDE and I can find an infinitesimal generator and its adjoint and I have an equation to solve.
But in the context of continuous-state and discrete-time, I don't know how to proceed. For example, in this reference: https://www.math.kth.se/matstat/gru/sf2955/metropolisB3.pdf With the example 2.5 (p.7) they found a gaussian invariant measure. But how to solve the equation 2.12 ?
Maybe, in this case it's a reversible Markov chain, so they compute $K(x,y)/K(y,x)$ and found the density $f$. But in general, if the Markov chain is not reversible, how to find $\pi$ ? Have you got some references and methods to do that ?