On return probabilities of random walks on compactly generated locally compact groups of polynomial growth

Abstract

We generalize Varopoulos' bound o return probabilities of random walks on compactly generated, unimodular groups of polynomial growth to arbitrary compactly generated, locally compact groups of polynomial growth. We review Varopoulos' proof of this bound, and show that the structure theory of groups of polynomial growth implies that in order to generalize Varopoulos' bound, t is sufficient to establish the invariance of probabilities of return under quasi-isometrics. Following Tessera and Coulhoun, we then prove the equivalence of large-scale Sobolev inequalities and bounds of probabilities of return, and show how large-scale Sobolev inequalities are invariant under quasi-isometrics.