Random walks on discrete groups of polynomial volume growth (Q1872275)

Assume that \(G\) is a finitely generated group of polynomial volume growth. This interesting article studies the asymptotic behavior of the convolution powers \(\mu ^n\) of a probability measure \(\mu\) supported on a finite generating subset of \(G\) and proves many interesting results. We quote the following main results. Upper and lower Gausisan estimates and Berry-Esseen estimates are proved and a central limit theorem is also proved. A parabolic inequality is proved, hence positive \(\mu\)-harmonic functions are constant. A characterization of \(\mu\)-harmonic functions that grow polynomially is also given.