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.
Publié le : 2007-04-14
Classification:
locally compact group,
Lie group,
amenable group,
random walk,
probability density,
heat kernel,
gaussian estimates,
convolution powers,
60B15,
60G50,
22E30,
22D05,
35B40
@article{1190831222,
author = {Dungey , Nick},
title = {Properties of centered random walks on locally
compact groups and Lie groups},
journal = {Rev. Mat. Iberoamericana},
volume = {23},
number = {1},
year = {2007},
pages = { 587-634},
language = {en},
url = {http://dml.mathdoc.fr/item/1190831222}
}
Dungey , Nick. Properties of centered random walks on locally
compact groups and Lie groups. Rev. Mat. Iberoamericana, Tome 23 (2007) no. 1, pp. 587-634. http://gdmltest.u-ga.fr/item/1190831222/