Properties of random walks on discrete groups: Time regularity and off-diagonal estimates

Abstract

In this paper we study some properties of the convolution powers K ( n ) = K ∗ K ∗ ⋯ ∗ K of a probability density K on a discrete group G, where K is not assumed to be symmetric. If K is centered, we show that the Markov operator T associated with K is analytic in L p ( G ) for 1 < p < ∞ , and prove Davies–Gaffney estimates in L 2 for the iterated operators T n . This enables us to obtain Gaussian upper bounds for the convolution powers K ( n ) . In case the group G is amenable, we discover that the analyticity and Davies–Gaffney estimates hold if and only if K is centered. We also estimate time and space differences, and use these to obtain a new proof of the Gaussian estimates with precise time decay in case G has polynomial volume growth.