Random walks on discrete groups of polynomial volume growth

Abstract

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.