Hostname: page-component-848d4c4894-hfldf Total loading time: 0 Render date: 2024-05-13T01:47:48.063Z Has data issue: false hasContentIssue false
  • English
  • Français

Isoperimetric Inequalities and Decay of Iterated Kernels for Almost-transitive Markov Chains

Published online by Cambridge University Press:  12 September 2008

L. Saloff-Coste
L. Saloff-Coste
Affiliation:
CNRS, Statistique et Probabilités, Université Paul Sabatier, France
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper gives precise isoperimetric inequalities for infinite graphs on which a group acts with finite quotient. Decay estimates are obtained for the iterated kernels of the associated random walks.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 4 , Issue 4 , December 1995 , pp. 419 - 442
Copyright
Copyright © Cambridge University Press 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Alexopoulos, G. (1992) A lower estimate for central probabilties on polycyclic groups. Canad. J. Math. 44 897–910.Google Scholar
[2]Bakry, D., Coulhon, Th., Ledoux, M. and Saloff-Coste, L. (1994) Sobolev inequalities in disguise. Preprint.CrossRefGoogle Scholar
[3]Babai, L. and Szegedy, M. (1992) Local expansion of symmetrical graphs. Comb. Prob. Comp. 1 1–11.CrossRefGoogle Scholar
[4]Carlen, E., Kusuoka, S. and Stroock, D. (1990) Upper bounds for symmetric Markov transition functions. Ann. Inst. H. Poincaré 26 245–287.Google Scholar
[5]Coulhon, Th. and Saloff-Coste, L. (1990) Puissances d'un opérateur régularisant. Ann. Inst. H. Poincaré 26 419–436.Google Scholar
[6]Coulhon, Th. and Saloff-Coste, L. (1990) Marches aléatoires non symétriques sur les groupes unimodulaires. C.R.A.S. Paris, 310 627–630.Google Scholar
[7]Coulhon, Th. and Saloff-Coste, L. (1993) Isopérimétrie pour les groupes et les variétés. Rev. Mat. Iberoamericana 9 293–314.CrossRefGoogle Scholar
[8]Diaconis, P. and Saloff-Coste, L. (1992) Nash's inequalities for finite Markov chains. J. Theor. Prob. (to appear).Google Scholar
[9]Diaconis, P. and Stroock, D. (1991) Geometric bounds for eigenvalues of Markov chains. Ann. Appl. Prob. 1 36–61.Google Scholar
[10]Fill, J. (1991) Eigenvalue bounds on convergence to stationarity for nonreversible Markov chains, with an application to the exclusion process. Ann. Appl. Prob. 1 62–87.Google Scholar
[11]Gromov, M. (1981) Groups of polynomial growth and expanding maps. I.H.E.S. Publ. Math. 53 53–78.CrossRefGoogle Scholar
[12]Hebisch, W. and Saloff-Coste, L. (1993) Gaussian estimates for Markov chains and random walks on groups. Ann. Prob. 21 673–709.Google Scholar
[13]Kaimanovich, V. (1992) Dirichlet norms, capacities and generalized isoperimetric inequalities for Markov operators. Potential Analysis 1 61–82.CrossRefGoogle Scholar
[14]Lust-Piquard, F. (1994) Lower bounds on ∥K n∥1→ ∞ for some contractions K of L 2(μ) with applications to Markov operators. Math. Ann. (to appear).Google Scholar
[15]Moser, J. (1961) On Harnack's theorem for elliptic differential equations. Comm. Pure Appl. Math. 14 577–591.CrossRefGoogle Scholar
[16]Pansu, P. (1989) Cohomologie L p des variétés à courbure négative, cas du degré 1. Rend. sem. Mat. Univ. Pol. Torino, Fascicolo Speciale, pp. 96–119.Google Scholar
[17]Pittet, Ch. (1993) Folner sequences in polycyclic groups. Rev. Mat. Iberoamericana (to appear).Google Scholar
[18]Robinson, D. (1992) Elliptic Operators on Lie Groups. Oxford University Press.Google Scholar
[19]Saloff-Coste, L. (1994) On global Sobolev inequalities. Forum Math. 6 271–286.CrossRefGoogle Scholar
[20]Soardi, P. and Woess, W. (1990) Amnability, unimodularity, and the spectral radius of random walks on infinite graphs. Math. Zeit. 205 471–486.Google Scholar
[21]Trofimov, V. (1985) Graphs with polynomial growth. Math. USSR Sb. 51 405–417.Google Scholar
[22]Varopoulos, N. (1984) Chaînes de Markov et inégalités isopérimétriques. C.R.A.S. Paris 298 233–236.Google Scholar
[23]Varopoulos, N. (1986) Théorie du potentiel sur des groupes et des variétés. C.R.A.S. Paris 302 203–205.Google Scholar
[24]Varopoulos, N. (1987) Convolution powers on locally compact groups. Bull. Sc. Math. 111 333–342.Google Scholar
[25]Varopoulos, N. (1991) Groups of superpolynomial growth. In Proc. ICM Satellite Conference on Harmonic Analysis, Springer-Verlag, Berlin.Google Scholar
[26]Varopoulos, N. (1990) Analysis and geometry on groups. In Proc. Kyoto ICM.Google Scholar
[27]Varopoulos, N., Saloff-Coste, L. and Coulhon, Th. (1993) Analysis and Geometry on Groups. Cambridge University Press, Cambridge.Google Scholar
[28]Woess, W. (1991) Topological groups and infinite graphs. Discrete Math. 95 373–384.Google Scholar
[29]Woess, W. (1994) Random walks on infinite graphs and groups – A survey on selected topics. Bull. L.M.S. 26 1–60.Google Scholar
[30]Zhou, X.Y. (1993) Green function estimates and their applications to the intersections of symmetric random walks. Stoch. Proc. Appl. 48 31–60.Google Scholar