  We extend the Dirichlet principle to non-reversible Markov processes on
countable state spaces. We present two variational formulas for the solution of
the Poisson equation or, equivalently, for the capacity between two disjoint
sets. As an application we prove a some recurrence theorems. In particular, we
show the recurrence of two-dimensional cycle random walks under a second moment
condition on the winding numbers.
