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

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.

### 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
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