Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-27T05:15:47.040Z Has data issue: false hasContentIssue false
  • English
  • Français

A Lower Estimate for Central Probabilities on Polycyclic Groups

Published online by Cambridge University Press:  20 November 2018

G. Alexopoulos
G. Alexopoulos*
Affiliation:
Université de Paris-Sud, Mathématiques, Bât. 425, 91405 Orsay Cedex, France
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We give a lower estimate for the central value μ*n(e) of the nth convolution power μ*···*μ of a symmetric probability measure μ on a polycyclic group G of exponential growth whose support is finite and generates G. We also give a similar large time diagonal estimate for the fundamendal solution of the equation (∂/∂t + L)u = 0, where L is a left invariant sub-Laplacian on a unimodular amenable Lie group G of exponential growth.

Keywords

31C0543A0560B15Polycyclic groupsvolume growthconvolution powerheat kernel
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 44 , Issue 5 , 01 October 1992 , pp. 897 - 910
Copyright
Copyright © Canadian Mathematical Society 1992

References

1. Avez, A., Harmonic functions on groups, Diff. Geom. and Relativity, (1976), 21-32.Google Scholar
2. Alexopoulos, G., On the mean distance of random walks on groups, Bull. Sci. Math. (2) 111(1987), 189–199.Google Scholar
3. Derrienic, Y., Quelques applications du théorème ergodique sous-additif Astérisque 74, 183–201.Google Scholar
4. Derrienic, Y.,Entropie, théorèmes limites et marches aléatoires, Lecture Notes in Math. 1210.Google Scholar
5. Grigorchuk, R.I., Degrees of growth of finitely generated groups and the theory of invariant means, Math. USSR, Izvestiya 25(1985).Google Scholar
6. Gromov, M., Groups of polynomial growth and expanding maps, Publications mathématiques de 1’ I.H.E.S. 53(1981).Google Scholar
7. Guivarc'h, Y., Lois de grands nombres et rayon spectrale d’ une march aléatoire sur un group de Lie, Astérisqu. 74(1980), 47–99.Google Scholar
8. Guivarc'h, Y., Croissance polynômiale et périodes des fonction harmoniques, Bull. Soc. Math. France, 101 (1973), 333–379.Google Scholar
9. Hörmander, L., Hypoelliptic second order differential operators, Acta Math. 119(1967), 147–171.Google Scholar
10. Kaimanovich, V.A., Brownian motion and harmonic functions on coverings of manifolds. An entropy approach, Soviet Math. Doklady (3) 33(1986), 812–816.Google Scholar
11. Milnor, J., Growth of finitely generated solvable groups, J. Diff. Geom. 72(1968), 447–449.Google Scholar
12. Raugi, A., Fonctions harmoniques sur les groupes localement compacts à base dénombrable, Bull. Soc. Math. France, Mémoire 54(1977), 5–118.Google Scholar
13. Ragunathan, M.S., Discrete subgroups of Lie groups, Springer-Verlag.Google Scholar
14. Shiryayev, A.N., Probability, Springer-Verlag, 1984.Google Scholar
15. Varadarajan, V.S., Lie groups, Lie algebras and their Representations, Springer-Verlag, 1984.Google Scholar
16. Th, N.. Varopoulos, Analysis on Lie groups, J. Funct. Analysis (2) 76(1988), 346–410.Google Scholar
17. Th, N.. Varopoulos, Information theory and harmonic functions, Bull. Sci. Math. (2e) 110(1986), 347–389.Google Scholar
18. Th, N.. Varopoulos, A potential theoritic property of solvable groups, Bull. Sci. Math. (2e) 108(1983), 263–273.Google Scholar
19. Th, N.. Varopoulos,Théorie du potentiel sur les groupes et les variétés, C.R. Acad. Sci. Paris (6) 3021 (1986), 203-205.Google Scholar
20. Th, N.. Varopoulos, Analysis and geometry on groups, Proceeding of the I.C.M., Kyoto, (1990). to appear.Google Scholar
21. Th, N.. Varopoulos, Groups of superpolynomial growth, preprint, 1990.Google Scholar
22. Vershik, A.M. and Kaimanovich, V.A., Random walks on discrete groups: Boundary and entropy, The Annals of Probability (3) 11(1983), 457–490.Google Scholar
23. Wolf, J.A., Growth of finitely generated solvable groups and curvature of Riemannian manifolds, J. Diff. Geom. 2(1968), 421–446.Google Scholar