Lower estimates for random walks on a class of amenable \(p\)-adic groups. (English) Zbl 1188.22005

Sami, Mustapha

Electron. J. Probab. 14, 1513-1531 (2009).

Let \(G\) be a locally compact group. The author investigates the growth of \(\varphi_n(e)\) for the densities \(\varphi_n\) of convolution powers \(\mu^n\) of symmetric probability measures \(\mu\in M^1(G)\) with a Haar density \(\varphi\) satisfying certain regularity conditions. The main result of the paper is a lower bound of the kind \(\varphi_n(e)\geq C/n^{1+\varepsilon}\) for some (explicit) constant \(\varepsilon=\varepsilon(G)\) for amenable algebraic groups over the \(p\)-adic numbers fields. Moreover, for metabelian \(p\)-adic groups, also an upper bound of the same kind is derived.

Reviewer: Michael Voit (Dortmund)

MSC:

22E35	Analysis on \(p\)-adic Lie groups
60B15	Probability measures on groups or semigroups, Fourier transforms, factorization
60G50	Sums of independent random variables; random walks

Keywords:

symmetric random walks; \(p\)-adic groups; estimates for transition probabilities