Green function estimates and their applications to the intersections of symmetric random walks

In this paper, a Harnack inequality for some difference operators arising from uniform symmetric random walks (see Definition 1.1 below) on is proved, and a criterion on the intersections of two independent random walks on graphs is derived. As applications, we obtain reasonable estimates for the Green functions of uniform symmetric random walks on . We also prove that the intersections of two independent uniform symmetric random walks on can occur infinitely often with probability one if d[greater-or-equal, slanted]4 and with probability zero if d[greater-or-equal, slanted]5, as for simple random walks on .