Harnack Inequalities of Hitting Distributions of Projections of Planar   Symmetric Random Walks on the Lattice Torus
  We give Harnack inequalities for the hitting distributions of a large family of symmetric random walks on $\Z^2$, and their projections onto the lattice torus $\Z^2_K$. This extends a framework for the simple random walk in Dembo, et al. (2006), and generalizes the results in Bass, Rosen (2007) to the toral projection. 
Michael Carlisle
math.PR