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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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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DOI: https://doi.org/10.1007/s004400200212
Keywords
- Volume Growth
- Polynomial Volume Growth