A variation principle for general Markov chain

We first recall some Dirichlet principles for Markov processes, especially for non-symmetric diffusion processes and Markov chains.  We will then show how this classical method can work for a (non-symmetric) Markov chain Pon a general state space, and obtain a Dirichlet principle for the solution of the general Poisson equation. The formula depends on the solution of the Poisson equation of P*, the dual Markov chain through an excessive measure. Some comparison theorems are presented.