Some Remarks on the Elliptic Harnack Inequality

Three short results are given concerning the elliptic Harnackinequality, in the context of random walks on graphs. The firstis that the elliptic Harnack inequality implies polynomial growthof the number of points in balls, and the second that the ellipticHarnack inequality is equivalent to an annulus-type Harnackinequality for Green's functions. The third result uses thelamplighter group to give a counter-example concerning the relationof coupling with the elliptic Harnack inequality. 2000 MathematicsSubject Classification 31B05 (primary), 60J35, 31C25 (secondary).