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 ${\mathbb{Z}}^2$, and their projections onto the lattice torus ${\mathbb{Z}}_K^2$. 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.