Revista Matemática Iberoamericana


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Volume 23, Issue 2, 2007, pp. 587–634
DOI: 10.4171/RMI/506

Published online: 2007-08-31

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

Nick Dungey[1]

(1) Macquarie University, Sydney, Australia

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 $1 < p < \infty$, 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.

Keywords: Locally compact group, Lie group, amenable group, random walk, probability density, heat kernel, Gaussian estimates, convolution powers

Dungey Nick: Properties of centered random walks on locally compact groups and Lie groups. Rev. Mat. Iberoamericana 23 (2007), 587-634. doi: 10.4171/RMI/506