A Dirichlet principle for non reversible Markov chains and some   recurrence theorems
  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. 
Alexandre Gaudilli\`ere and Claudio Landim
math.PR