Random walks on discrete groups of polynomial volume growth

• 成果类型: Article
• 署名作者: Alexopoulos, GK
• 署名单位: Universite Paris Saclay
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2002
• 页码: 723-801
• 关键词: CENTRAL-LIMIT-THEOREM; solvable lie-groups; harmonic-functions

摘要

Let it 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.