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 * ... * 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<p<\infty, 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.