Properties of centered random walks on locally compact groups and Lie groups

The basic aim of this paper is to study asymptotic properties of the convolution powers K^{(n)} = K*K* \cdots *K of a possibly non-symmetric probability density K on a locally compact, compactly generated group G . If K is centered, we show that the Markov operator T associated with K is analytic in L^p(G) for 1 , and establish Davies-Gaffney estimates in L^2 for the iterated operators T^n . These results enable us to obtain various Gaussian bounds on K^{(n)} . In particular, when G is a Lie group we recover and extend some estimates of Alexopoulos and of Varopoulos for convolution powers of centered densities and for the heat kernels of centered sublaplacians. Finally, in case G is amenable, we discover that the properties of analyticity or Davies-Gaffney estimates hold only if K is centered.