### abstract

- In a general setting we solve the following inverse problem: Given a positive operators $ R $, acting on measurable functions on a fixed measure space $(X,\mathcal B_X) $, we construct an associated Markov chain. Specifically, starting with a choice of $ R $(the transfer operator), and a probability measure $\mu_0 $ on $(X,\mathcal B_X) $, we then build an associated Markov chain $ T_0, T_1, T_2,\ldots $, with these random variables (rv) realized in a suitable probability space $(\Omega,\mathcal F,\mathbb P) $, and each rv taking values in $ X $, and with $ T_0 $ having the probability $\mu_0 $ as law. We further show how spectral data for $ R $, eg, the presence of $ R $-harmonic functions, propagate to the Markov chain. Conversely, in a general setting, we show that every Markov chain is determined by its transfer operator. In a range of examples we put this correspondence …