Open Access
August, 2007 Properties of centered random walks on locally compact groups and Lie groups
Nick Dungey
Rev. Mat. Iberoamericana 23(2): 587-634 (August, 2007).


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.


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Nick Dungey . "Properties of centered random walks on locally compact groups and Lie groups." Rev. Mat. Iberoamericana 23 (2) 587 - 634, August, 2007.


Published: August, 2007
First available in Project Euclid: 26 September 2007

zbMATH: 1130.60009
MathSciNet: MR2371438

Primary: 22D05 , 22E30 , 35B40 , 60B15 , 60G50

Keywords: ‎amenable group , Convolution powers , Gaussian estimates , heat kernel , Lie group , locally compact group , probability density , Random walk

Rights: Copyright © 2007 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.23 • No. 2 • August, 2007
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