## Lower bounds for random walks on amenable $$p$$-adic groups. (Bornes inférieures pour les marches aléatoires sur les groupes $$p$$-adiques moyennables.)(French)Zbl 1102.60041

Let $$G$$ be a locally compact, amenable group. Denote by $$e$$ the identity of $$G$$ and by $$d^rg$$ the right Haar measure. Suppose that $$d\mu(g)=\varphi(g)d^rg$$ is a symmetric probability measure on $$G$$, with $$\varphi(g)$$ a function with compact support. Denote by $$d\mu^{*n}(g)=d(\mu*\mu*\dotsb*\mu)(g)=\varphi_n(g)d^rg$$ the $$n$$th convolution power of $$d\mu(g)$$. One of the central problems in the study of the random walk described by the convolution powers of $$d\mu(g)$$, is to compute the rate of convergence in the limit $$\lim_{n\to\infty}\varphi(e)=0$$. For instance, if $$G$$ is an amenable, unimodular Lie group, one has the dicotomy $\varphi_n(e)\approx n^{-D/2}\Leftrightarrow \gamma(n)\approx n^D,\quad \varphi_n(e)\approx \exp(-n^{1/3})\Leftrightarrow \gamma(n)\approx \exp(n),$ where $$\gamma(n)$$ is the volume growth of $$G$$ [see N. Th. Varopoulos, L. Saloff-Coste and T. Coulhon, “Analysis on Lie groups” (1992; Zbl 0813.22003)]. In a more general situation, where $$G$$ is only a locally compact, compactly generated group of exponential growth, one has only the upper estimate $\varphi_n(e)\leq C\exp(-cn^{1/3}),\quad n\leq 1;$ see W. Hebisch and L. Saloff-Coste [Ann. Probab. 21, No. 2, 673–709 (1993; Zbl 0776.60086)]. Now let $$k$$ be a totally disconnected, non-discrete, locally compact field of characteristic zero, and let $$G$$ a connected, algebraic group over $$k$$. The paper examines the case in which $$G$$ is also compactly generated and amenable. Under the hypothesis that $$\varphi$$ is continuous and there exists a symmetric subset $$\Omega\subset G$$, containing $$e$$ and such that
$\inf\{\varphi(g):g\in \Omega\}>0,\quad G=\bigcup_{n\leq 0}\Omega^n,$ the author proves the following lower estimate: $\varphi_{2n}(e)\geq c\exp(-Cn^{1/3}),\quad n\geq 1.$ Under the additional hypothesis that $$G$$ is unimodular, there is also a lower estimate: $\frac{1}{C}\exp(-c_1n^{1/3})\leq\varphi_{2n}(e)\leq C\exp(-c_2n^{1/3}).$

### MSC:

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

### Keywords:

$$p$$-adic analysis

### Citations:

Zbl 0813.22003; Zbl 0776.60086
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