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Centered densities on Lie groups of polynomial volume growth
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  • Published: September 2002

Centered densities on Lie groups of polynomial volume growth

  • Georgios K. Alexopoulos1 

Probability Theory and Related Fields volume 124, pages 112–150 (2002)Cite this article

Abstract.

 We study the asymptotic behavior of the convolution powers of a centered density on a connected Lie group G of polynomial volume growth. The main tool is a Harnack inequality which is proved by using ideas from Homogenization theory and by adapting the method of Krylov and Safonov. Applying this inequality we prove that the positive -harmonic functions are constant. We also characterise the -harmonic functions which grow polynomially. We give Gaussian estimates for , as well as for the differences and . We give estimates, similar to the ones given by the classical Berry-Esseen theorem, for and . We use these estimates to study the associated Riesz transforms.

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Authors and Affiliations

  1. Laboratoire de Mathématique – UMR 8628, Université de Paris- Sud, Bât. 425, 91405 Orsay Cedex, France. e-mail: georges.alexopoulos@math.u-psud.fr, , , , , , FR

    Georgios K. Alexopoulos

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  1. Georgios K. Alexopoulos
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Received: 5 July 1999 / Revised version: 8 April 2002 / Published online: 22 August 2002

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Alexopoulos, G. Centered densities on Lie groups of polynomial volume growth. Probab Theory Relat Fields 124, 112–150 (2002). https://doi.org/10.1007/s004400200212

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  • Issue Date: September 2002

  • DOI: https://doi.org/10.1007/s004400200212

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Keywords

  • Volume Growth
  • Polynomial Volume Growth
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