The Annals of Probability

Random walks on discrete groups of polynomial volume growth

Georgios K. Alexopoulos

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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.

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Ann. Probab., Volume 30, Number 2 (2002), 723-801.

First available in Project Euclid: 7 June 2002

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Primary: 43A80: Analysis on other specific Lie groups [See also 22Exx] 60J15 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 22E25: Nilpotent and solvable Lie groups 22E30: Analysis on real and complex Lie groups [See also 33C80, 43-XX]

random walk group convolution Harnack inequality heat kernel


Alexopoulos, Georgios K. Random walks on discrete groups of polynomial volume growth. Ann. Probab. 30 (2002), no. 2, 723--801. doi:10.1214/aop/1023481007.

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