Let $\mu$ be a probability measure with finite support on a
discrete group $\Gamma$ of polynomial volume growth. The main purpose of this
paper is to study the asymptotic behavior of the convolution powers $\mu^{*n}$
of $\mu$. If $\mu$ is centered, then we prove upper and lower Gaussian
estimates. We prove a central limit theorem and we give a generalization of the
Berry–Esseen theorem. These results also extend to noncentered
probability measures. We study the associated Riesz transform operators. The
main tool is a parabolic Harnack inequality for centered probability measures
which is proved by using ideas from homogenization theory and by adapting the
method of Krylov and Safonov. This inequality implies that the positive
$\mu$-harmonic functions are constant. Finally we give a characterization of
the $\mu$-harmonic functions which grow polynomially.
Publié le : 2002-04-14
Classification:
random walk,
group,
convolution,
Harnack inequality,
heat kernel,
43A80,
60J15,
60B15,
20F65,
22E25,
22E30
@article{1023481007,
author = {Alexopoulos, Georgios K.},
title = {Random walks on discrete groups of polynomial volume
growth},
journal = {Ann. Probab.},
volume = {30},
number = {1},
year = {2002},
pages = { 723-801},
language = {en},
url = {http://dml.mathdoc.fr/item/1023481007}
}
Alexopoulos, Georgios K. Random walks on discrete groups of polynomial volume
growth. Ann. Probab., Tome 30 (2002) no. 1, pp. 723-801. http://gdmltest.u-ga.fr/item/1023481007/