no. 6
A geometric approach to on-diagonal heat kernel lower bounds on groups
[Une approche géométrique aux bornes inférieures sur la diagonale du noyau de la chaleur sur les groupes]
Coulhon, Thierry1 ; Grigor'yan, Alexander2 ; Pittet, Christophe3
1 Université de Cergy-Pontoise, Département de Mathématiques, 2 avenue Adolphe Chauvin, 95032 Cergy Cedex (France)
2 Imperial College, London DW7 2BZ (Grande-Bretagne)
3 Université Paul Sabatier, Laboratoire Émile Picard, 118 route de Narbonne, 31062 Toulouse Cedex (France)
Annales de l'Institut Fourier, Tome 51 (2001) no. 6, pp. 1763-1827.

On introduit une nouvelle méthode pour minorer sur la diagonale les noyaux de la chaleur des groupes de Lie non-compacts et des groupes infinis de type fini. Cette méthode permet de retrouver les bornes inférieures optimales pour les groupes de Lie unimodulaires moyennables et pour certains groupes de type fini, parmi lesquels les groupes polycycliques. Elle permet aussi de donner une interprétation géométrique de ces résultats. On obtient des résultats nouveaux pour certains groupes discrets admettant une structure de produit semi-direct avec groupe quotient abélien ou nilpotent. Parmi ces groupes, on trouvera ceux des transformations affines de la droite réelle engendrés par la translation 〈x↦x+1〉 et une homothétie 〈x↦λx〉 avec λ algébrique. On trouvera aussi certains produits en couronnes, comme les groupes d’allumeurs de réverbères à base nilpotente.

We introduce a new method for obtaining heat kernel on-diagonal lower bounds on non- compact Lie groups and on infinite discrete groups. By using this method, we are able to recover the previously known results for unimodular amenable Lie groups as well as for certain classes of discrete groups including the polycyclic groups, and to give them a geometric interpretation. We also obtain new results for some discrete groups which admit the structure of a semi-direct product or of a wreath product. These include the two- generators groups of affine transformations of the real line 〈x↦x+1,x↦λx〉 with λ algebraic, as well as lamplighter groups with nilpotent base.

DOI : 10.5802/aif.1874
Classification : 58J35, 60G50, 22E30, 20E22
Keywords: heat kernels on manifolds, random walks on graphs, Følner sets, first eigenvalue for the Dirichlet problem, Lie groups, finitely generated groups
Mot clés : noyaux de la chaleur sur les varietés, marches aléatoires sur les graphes, ensembles de Følner, première valeur propre pour le problème de dirichlet, groupes de Lie, groupes finiment engendrés

Coulhon, Thierry 1 ; Grigor'yan, Alexander 2 ; Pittet, Christophe 3

1 Université de Cergy-Pontoise, Département de Mathématiques, 2 avenue Adolphe Chauvin, 95032 Cergy Cedex (France)
2 Imperial College, London DW7 2BZ (Grande-Bretagne)
3 Université Paul Sabatier, Laboratoire Émile Picard, 118 route de Narbonne, 31062 Toulouse Cedex (France) 
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Coulhon, Thierry; Grigor'yan, Alexander; Pittet, Christophe. A geometric approach to on-diagonal heat kernel lower bounds on groups. Annales de l'Institut Fourier, Tome 51 (2001) no. 6, pp. 1763-1827. doi : 10.5802/aif.1874. https://aif.centre-mersenne.org/articles/10.5802/aif.1874/

[1] G.K. Alexopoulos Fonctions harmoniques bornées sur les groupes résolubles, C.R. Acad. Sci. Paris, Volume 305 (1987), pp. 777-779 | MR | Zbl

[2] G.K. Alexopoulos A lower estimate for central probabilities on polycyclic groups, Can. J. Math., Volume 44 (1992) no. 5, pp. 897-910 | DOI | MR | Zbl

[3] M.T. Barlow; A. Perkins Symmetric Markov chains in ℤ d : how fast can they move?, Probab. Th. Rel. Fields, Volume 82 (1989), pp. 95-108 | DOI | MR | Zbl

[4] L. Bartholdi The growth of Grigorchuk's torsion group, Internat. Math. Res. Notices, Volume 20 (1998), pp. 1049-1054 | DOI | MR | Zbl

[5] H. Bass The degree of polynomial growth of finitely generated groups, Proc. London Math. Soc., Volume 25 (1972), pp. 603-614 | DOI | MR | Zbl

[6] J. Cheeger A lower bound for the smallest eigenvalue of the Laplacian, Problems in Analysis: A Symposium in honor of Salomon Bochner (1970), pp. 195-199 | Zbl

[7] F.R.K. Chung Spectral Graph Theory, CBMS (Regional Conference Series in Mathematics), Volume 92 (1996) | Zbl

[8] F.R.K. Chung; A. Grigor'yan; S.-T. Yau Higher eigenvalues and isoperimetric inequalities on Riemannian manifolds and graphs, Comm. Anal. Geom., Volume 8 (2000) no. 5, pp. 969-1026 | MR | Zbl

[9] T. Coulhon Ultracontractivity and Nash type inequalities, J. Funct. Anal., Volume 141 (1996), pp. 510-539 | DOI | MR | Zbl

[10] T. Coulhon Large time behaviour of heat kernels on Riemannian manifolds: fast and slow decays, Journées équations aux dérivées partielles, St-Jean-de-Monts, Volume II,1-II,12 (1998) | Zbl

[11] T. Coulhon; A. Grigor'yan On diagonal lower bounds for heat kernels on non-compact manifolds and Markov chains, Duke Math. J., Volume 89 (1997) no. 1, pp. 133-199 | DOI | MR | Zbl

[12] T. Coulhon; A. Grigor'yan Random walks on graphs with regular volume growth, Geom. and Funct. Analysis, Volume 8 (1998), pp. 656-701 | DOI | MR | Zbl

[13] T. Coulhon; L. Saloff-Coste Isopérimétrie pour les groupes et les variétés, Rev. Mat. Iberoamericana, Volume 9 (1993) no. 2, pp. 293-314 | DOI | MR | Zbl

[14] E.B. Davies Heat kernels and spectral theory, Cambridge University Press, 1989 | MR | Zbl

[15] J. Dodziuk Difference equations, isoperimetric inequalities and transience of certain random walks, Trans. Amer. Math. Soc., Volume 284 (1984), pp. 787-794 | DOI | MR | Zbl

[16] A. Grigor'yan Heat kernel upper bounds on a complete non-compact manifold, Rev. Mat. Iberoamericana, Volume 10 (1994) no. 2, pp. 395-452 | DOI | MR | Zbl

[17] A. Grigor'yan Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc., Volume 36 (1999), pp. 135-249 | DOI | MR | Zbl

[18] A. Grigor'yan; M. Kelbert On Hardy-Littlewood inequality for Brownian motion on Riemannian manifolds, J. London Math. Soc. (2), Volume 62 (2000), pp. 625-639 | DOI | MR | Zbl

[19] R.I. Grigorchuk Degrees of growth of finitely generated groups and the theory of invariant means, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 49 (1984) no. 5, pp. 939-985 | MR | Zbl

[19] R.I. Grigorchuk Degrees of growth of finitely generated groups and the theory of invariant means, Math. USSR-Izv. (English transl.), Volume 25 (1985), pp. 259-300 | DOI | MR | Zbl

[20] M. Gromov Groups of polynomial growth and expanding maps, Publ. Math. I.H.E.S., Volume 53 (1981), pp. 53-73 | Numdam | MR | Zbl

[21] Y. Guivarc'h Croissance polynomiale et période des fonctions harmoniques, Bull. Soc. Math. France, Volume 101 (1973), pp. 333-379 | Numdam | MR | Zbl

[22] W. Hebisch On heat kernels on Lie groups, Math. Zeit., Volume 210 (1992), pp. 593-605 | DOI | MR | Zbl

[23] W. Hebisch; L. Saloff-Coste Gaussian estimates for Markov chains and random walks on groups, Ann. Prob., Volume 21 (1993), pp. 673-709 | DOI | MR | Zbl

[24] J. Jenkins Growth of connected locally compact groups, J. Funct. Anal., Volume 12 (1973), pp. 113-127 | DOI | MR | Zbl

[25] V.A. Kaimanovich; A.M. Vershik Random walks on discrete groups: boundary and entropy, Ann. Prob., Volume 11 (1983) no. 3, pp. 457-490 | DOI | MR | Zbl

[26] H. Kesten Symmetric random walks on groups, Trans. Amer. Math. Soc., Volume 92 (1959), pp. 336-354 | DOI | MR | Zbl

[27] F. Lust-Piquard Lower bounds on ∥K n ∥ 1→∞ for some contractions K of L 2 (μ), with some applications to Markov operators, Math. Ann., Volume 303 (1995), pp. 699-712 | DOI | MR | Zbl

[28] V.G. Maz'ya Sobolev spaces, Izdat. Leningrad Gos. Univ. Leningrad, Springer, 1985 | MR | Zbl

[29] G.D. Mostow On the fundamental group of a homogeneous space, Ann. Math., Volume 66 (1957) no. 2, pp. 249-255 | DOI | MR | Zbl

[30] Ch. Pittet Følner sequences on polycyclic groups, Rev. Mat. Iberoamericana, Volume 11 (1995) no. 3, pp. 675-686 | DOI | MR | Zbl

[31] Ch. Pittet The isoperimetric profile of homogeneous Riemannian manifolds, J. Diff. Geom., Volume 54 (2000) no. 2, pp. 255-302 | MR | Zbl

[32] Ch. Pittet; L. Saloff-Coste A survey on the relationship between volume growth, isoperimetry, and the behavior of simple random walk on Cayley graphs, with examples (1997) (Preprint)

[33] Ch. Pittet; L. Saloff-Coste; eds. J. Cossey, C.F. Miller III Amenable groups, isoperimetric profiles and random walks, Geometric group theory down under. Proceedings of a special year in geometric group theory, Canberra, Australia, 1996 (1999) | Zbl

[34] Ch. Pittet; L. Saloff-Coste On the stability of the behavior of random walks on groups, J. Geom. Anal., Volume 10 (2000) no. 4, pp. 713-737 | DOI | MR | Zbl

[35] Ch. Pittet; L. Saloff-Coste On random walks on wreath products (To appear in Annals Proba) | Zbl

[36] M.S. Raghunathan Discrete subgroups of Lie groups, Ergebnisse der Mathematik, 68, Springer, Berlin, 1972 | MR | Zbl

[37] D.J.S. Robinson A course in the theory of groups, Graduate texts in Mathematics, Springer, 1993 | MR | Zbl

[38] D. Segal Polycyclic groups, Cambridge Tracts in Mathematics, Volume 82 (1983) | MR | Zbl

[39] D.W. Stroock; eds. L.H.Y. Chen, K.P. Choi Estimates on the heat kernel for the second order divergence form operators, Probability theory. Proceedings of the 1989 Singapore Probability Conference held at the National University of Singapore, June 8-16 1989 (1992), pp. 29-44 | Zbl

[40] J. Tits Appendix to Gromov M., Groups of polynomial growth and expanding maps, Publ. Math.I.H.E.S., Volume 53 (1981), pp. 74-78 | Numdam | MR | Zbl

[41] N.Th. Varopoulos A potential theoretic property of soluble groups, Bull. Sci. Math., 2e série, Volume 108 (1983), pp. 263-273 | MR | Zbl

[42] N.Th. Varopoulos Random walks on soluble groups, Bull. Sc. Math., 2e série, Volume 107 (1983), pp. 337-344 | MR | Zbl

[43] N.Th. Varopoulos Convolution powers on locally compact groups, Bull. Sc. Math., 2e série, Volume 111 (1987), pp. 333-342 | MR | Zbl

[44] N.Th. Varopoulos Analysis on Lie groups, J. Funct. Anal., Volume 76 (1988), pp. 346-410 | DOI | MR | Zbl

[45] N.Th. Varopoulos Groups of superpolynomial growth, Proceedings of the ICM satellite conference on Harmonic analysis (1991) | Zbl

[46] N.Th. Varopoulos Diffusion on Lie groups II, Can. J. Math., Volume 46 (1994) no. 5, pp. 1073-1092 | DOI | MR | Zbl

[47] N.Th. Varopoulos; L. Saloff-Coste; T. Coulhon Analysis and geometry on groups, Cambridge University Press, Cambridge, 1992 | MR | Zbl

[48] W. Woess Random walks on infinite graphs and groups, Cambridge Tracts in Mathematics, 138, Cambridge Univ. Press, 2000 | MR | Zbl

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