Random walks on unimodular \(p\)-adic groups. (English) Zbl 1076.60035

Mustapha, Sami

Stochastic Processes Appl. 115, No. 6, 927-937 (2005).

Let \(\mathbb{Q}_{p}\) be the \(p\)-adic field and \(k , l \geq 0\) be integers. Define \(G = \mathbb{Q}_{p}^{k} \ltimes_{\sigma} (\mathbb{Q}^{ast})^{l}\), where the multiplicative group

•  Encyclopedia of Mathematics
•  nLab
•  Wikipedia
•  Wolfram MathWorld

\((\mathbb{Q}^{\ast})^{l}\) acts on the vector space \(\mathbb{Q}^{k}\), \(\sigma(y) = (\chi_{1}(y),\dots, \chi_{k}(y)),\) and \(\chi_{i}\) are morphisms \((\mathbb{Q}^{\ast})^{l} \rightarrow \mathbb{Q}^{\ast}.\) Let \(\mu\) be a probability measure on \(G\) with symmetric density \(\varphi(g)\), compact support or a fast decay at infinity. We consider the \(G\)-valued symmetric random walk with law \(\mu\). Then the return probability to \(e\), the identity of \(G\), is given by \(\varphi_{2n}(e)\), where \(d\mu^{\ast n}(g) = \varphi_{n}(g)dg.\) The following inequality is proved: \[ \tfrac{1}{C} \exp(- c_{1} n^{1/3}) \leq \varphi_{2n}(e) \leq C\exp(- c_{2} n^{1/3}), \] where constants \(c_{1}, c_{2}, C > 0\) are independent of \(n\).

Reviewer: Yasuhiro Asoo (Okayama)

MSC:

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

Keywords:

recurrent probabilty

References:

[1]	Alexopoulos, G., A lower estimate for central probabilities on polycyclic groups, Can. J. Math., 44, 897-910, 1992 · Zbl 0762.31003
[2]	Bachman, G., Introduction to p-Adic Numbers and Valuation Theory, 1964, Academic Press: Academic Press New York · Zbl 0192.40103
[3]	Borel, A.; Tits, J., Groupes réductifs, Publications Math. IHES, 27, 55-150, 1965
[4]	Cassels, J. W.S., Local Fields, 1986, Cambridge University Press: Cambridge University Press Cambridge · Zbl 0595.12006
[5]	Coulhon, Th.; Grigor’yan, A.; Pittet, Ch., A geometric approach to on-diagonal heat kernel lower bounds on groups, Ann. Inst. Fourier, 51, 6, 1763-1827, 2001 · Zbl 1137.58307
[6]	Guivarc’h, Y., Croissance polynomiale et périodes des fonctions harmoniques, Bull. Soc. Math. France, 101, 333-379, 1973 · Zbl 0294.43003
[7]	Hebisch, W.; Saloff-Coste, L., Gaussian estimates for Markov chains and random walks on groups, Ann. Probab., 21, 673-709, 1993 · Zbl 0776.60086
[8]	Ch. Pittet, L. Saloff-Coste, Amenable Groups, Isoperimetric Profiles and Random Walks, Geometric Group Theory Down under (Canberra, 1996), de Gruyter, Berlin, 1999, pp. 293–316. · Zbl 0934.43001
[9]	Pittet, Ch.; Saloff-Coste, L., On random walks on wreath products, Ann. Probab., 30, 2, 1-30, 2002 · Zbl 1021.60004
[10]	Pittet, Ch.; Saloff-Coste, L., Random walks on finite rank solvable groups, J. Eur. Math. Soc., 4, 313-342, 2003 · Zbl 1057.20026
[11]	Raja, C. R.E., On classes of p-adic Lie groups, New York J. Math., 5, 101-105, 1999 · Zbl 0923.22006
[12]	Taibleson, M. H., Fourier Analysis on Local Fields, 1975, Princeton University Press, University of Tokyo Press: Princeton University Press, University of Tokyo Press Princeton, NJ, Tokyo · Zbl 0319.42011
[13]	Varopoulos, N. Th., Random walks on soluble groups, Bull. Sci. Math., 107, 337-344, 1983 · Zbl 0532.60009
[14]	Varopoulos, N. Th., A potential theoretic property of soluble groups, Bull. Sci. Math., 108, 263-273, 1983 · Zbl 0546.60008
[15]	N.Th. Varopoulos, Groups of superpolynomial growth, in: S. Igasi (Ed.), Harmonic Analysis, ICM Satellite Conference Proceedings, Sendai, 1990, Springer, Berlin, 1991. · Zbl 0802.43002
[16]	Varopoulos, N. Th.; Saloff-Coste, L.; Coulhon, Th., Analysis and geometry on groups
[17]	Varopoulos, N. Th., Analysis on Lie groups, Rev. Mat. Iberoamericana, 12, 3, 791-916, 1996 · Zbl 0881.22009