The basic aim of this paper is to study asymptotic properties of the convolution powers of a possibly non-symmetric probability density on a locally compact, compactly generated group . If is centered, we show that the Markov operator associated with is analytic in for , and establish Davies-Gaffney estimates in for the iterated operators . These results enable us to obtain various Gaussian bounds on . In particular, when 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 is amenable, we discover that the properties of analyticity or Davies-Gaffney estimates hold only if is centered.
Cite this article
Nick Dungey, Properties of centered random walks on locally compact groups and Lie groups. Rev. Mat. Iberoam. 23 (2007), no. 2, pp. 587–634DOI 10.4171/RMI/506