Random walks on graphs with regular volume growth

Geometric and Functional Analysis

Volumn 8, Issue 4, 1998, Pages 656-701

  Coulhon, Thierry ^a   Grigoryan, Alexander ^b  

^a UNIVERSITÉ DE CERGY PONTOISE   (France)

^b IMPERIAL COLLEGE LONDON   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0032382061     PISSN: 1016443X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s000390050070     Document Type: Article

References (43)

1. D. BAKRY, T. COULHON, M. LEDOUX, L. SALOFF-COSTE, Sobolev inequalities in disguise, Indiana Univ. Math. J. 124:4 (1995), 1033-1074.
2. I. BENJAMINI, I. CHAVEL, E.A. FELDMAN, Heat kernel lower bounds on Riemannian manifolds using the old ideas of Nash, Proceedings of London Math. Soc. 72 (1996), 215-240.
3. E.A. CARLEN, S. KUSUOKA, D.W. STROOCK, Upper bounds for symmetric Markov transition functions, Ann. Inst. H. Poincaré, proba. et stat. (1987), suppl. au no 2, 245-287.
4. K. CARNE, A transmutation formula for Markov chains, Bull. Sc. Math. 2ème série 109 (1985), 399-403.
5. G. CARRON, Inégalités isopérimétriques de Faber-Krahn et conséquences, in: Actes de la table ronde de géométrie différentielle en l'honneur de Marcel Berger, Collection SMF séminaires et congrès 1 1994.
6. F.R.K. CHUNG, Spectral Graph Theory, CBMS Lecture Notes 92, AMS Publications, 1995.
7. T. COULHON, Espaces de Lipschitz et inégalités de Poincaré, J. Funct. Anal. 136 :1 (1996), 81-113.
8. T. COULHON, Ultracontractivity and Nash type inequalities, J. Funct. Anal. 141 (1996), 510-539.
9. T. COULHON, A. GRIGOR'YAN, On-diagonal lower bounds for heat kernels on non-compact manifolds and Markov chains, Duke Math. J. 89 (1997), 133-199.
10. T. COULHON, M. LEDOUX, Isopérimétrie, décroissance du noyau de la chaleur et transformations de Riesz: un contre-exemple. Ark. Mat. 32 (1994), 63-77.
11. T. COULHON, L. SALOFF-COSTE, Puissances d'un opérateur régularisant, Ann. Inst. Henri Poincaré, proba. et stat. 26:3 (1990). 419-436.
12. T. COULHON, L. SALOFF-COSTE, Marches aléatoires non symétriques sur les groupes unimodulaires, C.R. Acad. Sci. Paris, Sér. I 310 (1990), 627-630.
13. T. COULHON, L. SALOFF-COSTE, Isopérimétrie pour les groupes et les variétés, Revista Mathemática Iberoamericana 9:2 (1993), 293-314.
14. T. COULHON, L. SALOFF-COSTE, Minorations pour les chaînes de Markov unidimensionnelles, Probab. Theory Relat. Fields 97 (1993), 423-431.
15. T. COULHON, L. SALOFF-COSTE, Variétés riemanniennes isométriques à l'infini, Revista Mathemática Iberoamericana 11:3 (1995), 687-726.
16. E.B. DAVIES, Heat kernel bounds, conservation of probability and the Feller property, J. d'Analyse Math. 58 (1992), 99-119.
17. E.B. DAVIES, Large deviations for heat kernels on graphs, J. London Math. Soc. (2) 47 (1993), 65-72.
18. T. DELMOTTE, Inégalité de Harnack elliptique sur les graphes, Colloq. Math 72:1 (1997), 19-37.
19. T. DELMOTTE, Parabolic Harnack inequality and estimates of Markov chains on graphs, preprint.
20. E. DE GIORGI, Sulla differenziabilita e l'analiticita delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Nat., ser. 3, 3 (1957), 25-43.
21. A. GRIGOR'YAN, The heat equation on non-compact Riemannian manifolds, (in Russian) Matem. Sbornik 182:1 (1991), 55-87. Engl. transl., Math. USSR Sb. 72:1 (1992), 47-77.
22. A. GRIGOR'YAN, The heat equation on non-compact Riemannian manifolds, (in Russian) Matem. Sbornik 182:1 (1991), 55-87. Engl. transl., Math. USSR Sb. 72:1 (1992), 47-77.
23. A. GRIGOR'YAN, Heat kernel upper bounds on a complete non-compact manifold, Revista Mathemática Iberoamericana 10:2 (1994), 395-452.
24. A. GRIGOR'YAN, Integral maximum principle and its applications, Proc. Roy. Soc. Edinburgh 124A (1994), 353-362.
25. A. GRIGOR'YAN, Gaussian upper bounds for the heat kernel on arbitrary manifolds, J. Diff. Geom. 45 (1997), 33-52.
26. A.K. GUSHCHIN, On the uniform stabilization of solutions of the second mixed problem for a parabolic equation, (in Russian) Matem. Sbornik 119(161): 4 (1982), 451-508. Engl. transl., Math. USSR Sb. 47 (1984), 439-498.
27. A.K. GUSHCHIN, On the uniform stabilization of solutions of the second mixed problem for a parabolic equation, (in Russian) Matem. Sbornik 119(161): 4 (1982), 451-508. Engl. transl., Math. USSR Sb. 47 (1984), 439-498.
28. A.K. GUSHCHIN, On the uniform stabilization of solutions of the second mixed problem for a parabolic equation, (in Russian) Matem. Sbornik 119(161): 4 (1982), 451-508. Engl. transl., Math. USSR Sb. 47 (1984), 439-498.
29. W. HEBISH, L. SALOFF-COSTE, Gaussian estimates for Markov chains and random walks on groups, Ann. Probab. 21 (1993), 673-709.
30. J. HEINONEN, P. KOSKELA, Quasiconformal maps in metric spaces with controlled geometry, preprint.
31. O.A. LADYZENSKAJA, V.A. SOLONNIKOV, N.N. URAL'CEVA, Linear and Quasilinear Equations of Parabolic Type, Providence, Rhode Island, 1968.
32. p and mean value properties of subharmonic functions on Riemannian manifolds, Acta Mathematica 153 (1984), 279-301.
33. P. LI, S.-T. YAU, On the parabolic kernel of the Schrödinger operator, Acta Math. 156:3-4 (1986), 153-201.
34. 2(μ), with some applications to Markov operators, Math. Ann. 303 (1995), 699-712.
35. J. MOSER, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101-134.
36. M.M.H. PANG, Heat kernels on graphs, J. London Math. Soc. II, 47:1 (1993), 50-64.
37. E. RUSS, Temporal regularity for random walks and Riesz transforms on graphs for 1 ≤ p ≤ 2, preprint.
38. L. SALOFF-COSTE, Uniformly elliptic operators on Riemannian manifolds, J. Diff. Geom. 36 (1992), 417-450.
39. L. SALOFF-COSTE, A note on Poincaré, Sobolev, and Harnack inequalities, Duke Math J., I.M.R.N. 2 (1992), 27-38.
40. D.W. STROOCK, Estimates on the heat kernel for the second order divergence form operators, in: Probability Theory, Proceedings of the 1989 Singapore Probability Conference, National University of Singapore, June 8-16, 1989 (L.H.Y. Chen, K.P. Choi, K. Hu, J.H. Lou, eds.) Walter De Gruyter, (1992), 29-44.
41. N.TH. VAROPOULOS, Long range estimates for Markov chains, Bull. Sc. Math., 2ème série 109 (1985), 113-119.
42. N.TH. VAROPOULOS, Isoperimetric inequalities and Markov chains, J. Funct. Anal. 63 (1985), 215-239.
43. N.TH. VAROPOULOS, Analysis on Lie groups, J. Funct. Anal. 76 (1988), 346-410.