Theorie du Potentiel sur les Graphes et les Varietes

  • A. Ancona
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Part of the Lecture Notes in Mathematics book series (LNM, volume 1427)

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Références Bibliographiques

  1. [Aik]
    H. Aikawa, On the thinness in a Lipschitz domain, Analysis, 5,1985, 345–382.MathSciNetCrossRefMATHGoogle Scholar
  2. [Ale]
    G. Alexopoulos, Fonctions harmoniques bornées sur les groupes résolubles, C.R.Acad.Sci. Paris, 305, 1987, 777–779MathSciNetMATHGoogle Scholar
  3. [Anc.1]
    A. Ancona, Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien, Ann Inst.Fourier, XVIII, 4, 1978, 169–213MathSciNetCrossRefMATHGoogle Scholar
  4. [Anc.2]
    A. Ancona, Une propriété de la compactification de Martin d'un domaine euclidien, Ann. Inst. Fourier, XIX, 4, 1979, 71–90.MathSciNetCrossRefMATHGoogle Scholar
  5. [Anc.3]
    A. Ancona, Negatively curved manifolds,elliptic operators and the Martin Boundary, Annals of Maths, 125, 1987, 495–536.MathSciNetCrossRefMATHGoogle Scholar
  6. [Anc.4]
    A. Ancona, Positive harmonic functions and hyperbolicity, Potential Theory, Prague 1987, Springer Lecture notes no1344, 1–23.Google Scholar
  7. [Anc.5]
    A. Ancona, Régularité d'accès des bouts et frontière de Martin des domaines euclidiens, J. Math. Pures et Àp., 63, 1984,215–260.MathSciNetMATHGoogle Scholar
  8. [And]
    M.T. Anderson, The Dirichlet problem at infinity for manifolds of negative curvature, J. Diff. Geo. 18, 1983, 701–721.MathSciNetMATHGoogle Scholar
  9. [A.S]
    M.T. Anderson and R. Schoen, Positive harmonic functions on complete manifolds of negative curvature, Ann. of Maths 121, 1985, 429–461.MathSciNetCrossRefMATHGoogle Scholar
  10. [Av]
    A. Avez, Harmonic functions on groups, Differential Geometry and relativity, Reidel, 1976, 27–32.Google Scholar
  11. [Az]
    R. Azencott, Behavior of diffusion semi-groups at infinity, Bull. Soc. Math. France, 102, 1974, 193–240.MathSciNetMATHGoogle Scholar
  12. [BLP]
    P. Baldi, N. Lohoué, J. Peyrière, Sur la classification des groupes récurrents, C.R.A.S. Paris, t. 285, 1103–1104, 1977.MATHGoogle Scholar
  13. [Ba]
    H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, Proc. Lond. Math. Soc., 25, 1972, 603–614.MathSciNetCrossRefMATHGoogle Scholar
  14. [Bau]
    H. Bauer, Harmonishe raume und ihre Potential Theorie, Lecture Notes 22, 1966. Springer.Google Scholar
  15. [Ben]
    M. Benedicks, Positive harmonic functions vanishing on the boundary of certain domains of Rn, Ark. för Math. 18, 1, 1980,53–72.MathSciNetCrossRefMATHGoogle Scholar
  16. [B.D.]
    A. Beurling, J. Deny, Dirichlet spaces, Proc. Nat. Acad. Sc., 45, 208–215, 1959.MathSciNetCrossRefMATHGoogle Scholar
  17. [B.G]
    R. M. Blumenthal and R. K. Getoor, Markov processes and Potential theory, 1968, Academic Press, New-York and London.MATHGoogle Scholar
  18. [Bre1]
    M. Brelot, Axiomatique des fonctions harmoniques, Les Presses de l'Université de Montréal, 1969.Google Scholar
  19. [Bre2]
    M. Brelot, Sur le principe des singularités positives et la topologie de R.S.Martin, Ann. Univ. Grenoble, XIII,23, 1947–48,113–142.MathSciNetMATHGoogle Scholar
  20. [Br.Do]
    M. Brelot et J.L. Doob, Limites angulaires et limites fines, Ann. Inst. Fourier 13, 2, 1963, 395–415.MathSciNetCrossRefMATHGoogle Scholar
  21. [Broo]
    R. Brooks, The fundamental group and the spectrum of the laplacian, Comm. Math. Helv.56, 1981, 581–598.MathSciNetCrossRefMATHGoogle Scholar
  22. [Cha]
    I. Chavel, Eigenvalues in Riemannian geometry, Academic Press Inc., 1984Google Scholar
  23. [C.E]
    J. Cheeger and D. Ebin, Comparison theorems in Differential geometry, North Holland Publ. Co, Amsterdam, 1975MATHGoogle Scholar
  24. [C.G.T]
    J. Cheeger, M. Gromov, M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator and the geometry of complete manifolds, J. Diff. Geo., 17, 1983, 15–53.MathSciNetMATHGoogle Scholar
  25. [C.Y]
    S.Y. Cheng and S.T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math., XXVIII, 1975, 333–354.MathSciNetCrossRefMATHGoogle Scholar
  26. [C.L.Y]
    S.Y. Cheng, P. Li and S.T. Yau, On the upper estimate of the heat kernel of a complete Riemannian manifold, Amer. J. Math., 103, 1981, 1021–1063.MathSciNetCrossRefMATHGoogle Scholar
  27. [Cho]
    G. Choquet, Lectures on Analysis, tome 2, W. A. Benjamin Inc, 1969.Google Scholar
  28. [C.D]
    G. Choquet, J. Deny, Sur l'équation de convolution μ=μ*σ, C.R. Acad. Sci. 250,1960,799–801.MathSciNetMATHGoogle Scholar
  29. [C.C]
    C.Constantinescu, A.Cornea, Potential theory on harmonic spaces, 1972, Springer Verlag.Google Scholar
  30. [Dav]
    E. B. Davies, Heat kernels and spectral theory, Cambridge tracts in Mathematics, no92, 1989.Google Scholar
  31. [Den]
    J.Deny, Méthodes hilbertiennes en théorie du Potentiel, Cours de Stresa, C.I.M.E., Jul. 1969.Google Scholar
  32. [Der.1]
    Y. Derriennic, Lois "zéro ou deux" pour les processus de Markov. Applications aux marches aléatoires, Ann. Inst. Poincaré, XII,2, 1976, 111–129.MathSciNetMATHGoogle Scholar
  33. [Der.2]
    Y. Derriennic, Quelques applications du théorème ergodique sous-additif. Astérisque, 74, 183–201.Google Scholar
  34. [Doo.1]
    J. L. Doob, Classical Potential theory and its probabilistic counterpart, Springer Verlag, New-York, 1984.CrossRefMATHGoogle Scholar
  35. [Doo.2]
    J. L. Doob, Conditionned Brownian motion and the boundary limits of harmonic functions, Bull. Soc. Math. de France, 85, 1957,431–458.MathSciNetMATHGoogle Scholar
  36. [Dyn]
    E. Dynkin, Markov Processes, Springer Verlag, Vol. 1–2,New-York, 1965.Google Scholar
  37. [E.O]
    P. Eberlein and B. O'Neill, Visibility manifolds, Pac. J. of Math. 46, 1973, 45–109.MathSciNetCrossRefMATHGoogle Scholar
  38. [F.S]
    E. B. Fabes and D. W. Stroock, A new proof of Moser's parabolic Harnack inequality via the old ideas of Nash, Arch. Rat. Mech. Anal., 96, 327–338.Google Scholar
  39. [Fer]
    J.L. Fernandez, On the existence of Green's function in Riemannian manifolds, Proc of the A.M.S., 96,2, 1986, 284–286.CrossRefMATHGoogle Scholar
  40. [Fre]
    M. Freire, Positive harmonic functions on product of manifolds, Preprint.Google Scholar
  41. [Fri]
    A. Friedman, Partial Differential Equations of Parabolic type, Englewood Cliffs, NJ, Prentice Hall, 1964.MATHGoogle Scholar
  42. [Fu]
    M.Fukushima, Dirichlet forms ans Markov processes, North-Holland, 1980.Google Scholar
  43. [Fur]
    H. Furstenberg, A Poisson formula for semi-simple Lie Groups, Ann. of Math.77,2, 1963, 335–386.MathSciNetCrossRefMATHGoogle Scholar
  44. [G-T]
    D. Gildbarg and N. S. Trudinger, Elliptic partial differential equations of the second order,2nd edition, Springer, New-York,1983.CrossRefGoogle Scholar
  45. [Gow]
    K. Gowrisankaran, Fatou-Doob limit theorems in the axiomatic setting of Brelot, Ann. Inst. Fourrier,XVI, 2, 1966, 465–467.MathSciNetMATHGoogle Scholar
  46. [Gre]
    F.P. Greenleaf, Invariant means on topological groups, Van Nostrand, 1969.Google Scholar
  47. [Gr.1]
    M. Gromov, Hyperbolic groups, Essays in group theory, M.S.R.I Publications, 8, 1987, 75–263.MathSciNetCrossRefGoogle Scholar
  48. [Gr.2]
    M. Gromov, Groups of polynomial growth and expanding maps, Publications de l'I.H.E.S, 53,1981, 53–78.MathSciNetMATHGoogle Scholar
  49. [Gui.1]
    Y. Guivarc'h, Mouvement brownien sur les revêtements d'une variété compacte, C.R.Acad. Sc. Paris,892, 1981,851–853.MathSciNetMATHGoogle Scholar
  50. [Gui.2]
    Y. Guivarc'h, Sur la représentation intégrale des fonctions harmoniques et des fonctions propres positives dans un espace Riemannien symétrique, Preprint, Université de Rennes 1.Google Scholar
  51. [Gui.3]
    Y. Guivarc'h, Sur la loi des grands nombres et le rayon spectral d'une marche aléatoire, Astérisque 74, 1980 (Journées sur les marches aléatoires, Nancy 1979).Google Scholar
  52. [G.K.R]
    Y.Guivarc'h, M.Keane, B.Roynette, Marches aléatoires sur les groupes de Lie, Lecture Notes in Mathematics, 624, Springer Verlag.Google Scholar
  53. [Her]
    R. M. Hervé, Recherches sur la théorie des fonctions surharmoniques et du potentiel, Ann. Inst. Fourier, XII, 1962, 415–471.CrossRefMATHGoogle Scholar
  54. [H.W]
    R.A. Hunt, R.L. Wheeden, Positive Harmonic functions on Lipschitz doamins, Trans. Amer. Math. Soc., 147,1970, 507–527.MathSciNetCrossRefMATHGoogle Scholar
  55. [J.K]
    D. Jerison, C. Kenig, Boundary behavior of harmonic functions in non-tangentially accessible domains, Advances in Math.,46, 1982, 80–147.MathSciNetCrossRefMATHGoogle Scholar
  56. [Kai]
    V.A. Kaimanovich, Brownian motion and harmonic functions on covering manifolds. An entropy approach, Soviet Math. Dokl. 33,(3),1986, 812–816.Google Scholar
  57. [K.V]
    V. A. Kaimanovich and A. M. Vershik, Random walks on discrete groups: Boundary and entropy, Ann. Proba. 11, 3, 1983, 457–490.MathSciNetCrossRefMATHGoogle Scholar
  58. [K.L]
    L. Karp and P. Li, The heat equation on complete Riemannain manifolds. Pretirage.Google Scholar
  59. [Kif]
    Y. Kifer, Brownian motion and positive harmonic functions on complete manifolds of non positive curvature, Pitman Research Notes in Math. Series, 150, 1986, 187–232.MathSciNetMATHGoogle Scholar
  60. [K-L]
    Y.Kifer, F.Ledrappier, Hausdorff dimension of harmonic measures on negatively curved manifolds, à paraitre in Trans. Amer. Math. Soc.Google Scholar
  61. [K.T]
    A. Koranyi and J. Taylor, Minimal solutions of the heat equation and uniqueness of the positive Cauchy problem on homogeneous spaces, Proc.Amer.Math.Soc., 94, 1985, 273–278.MathSciNetCrossRefMATHGoogle Scholar
  62. [K.W]
    H.Kunita, T.Wanatabe, Markov processes and Martin boundaries, Ill. J. Math., 9, 1965.Google Scholar
  63. [Led.1]
    F. Ledrappier, Propriété de Poisson et courbure négative, C.R.Acad.Sci. Paris, 305, 1987, 191–194.MathSciNetMATHGoogle Scholar
  64. [Led.2]
    F. Ledrappier, Ergodic properties of brownian motion on covers of compact negatively curved manifolds, Bol. Soc. Bras. Mat.,19,1988, 115–140.MathSciNetCrossRefMATHGoogle Scholar
  65. [Lel]
    J. Lelong-Ferrand, Etude au voisinage d'un point frontière des fonctions surharmoniques positives dans un demi-espace, Ann. Sc. Ec. Norm. Sup., 66, 1949, 125–159.MathSciNetMATHGoogle Scholar
  66. [L.Y]
    P. Li and S. T. Yau, On the parabolic kernel of the Schrodinger operator, Acta Math., 156, 1986, 153–201.MathSciNetCrossRefGoogle Scholar
  67. [Lyo 1]
    T.J. Lyons,Instability of the Liouville property for quasi-isometric Riemannian manifolds and reversible Markov chains, J. Diff. Geo. 26, 1987, 33–66.MathSciNetMATHGoogle Scholar
  68. [Lyo 2]
    T.J. Lyons, A simple criterion for transience of a reversible markov chain, Ann. Proba. 11, 1983, 393–402.MathSciNetCrossRefMATHGoogle Scholar
  69. [L.M]
    T. Lyons and H. P. McKean, Winding of Plane brownian motion, Advances in Math., 51, 1984, 212, 225.MathSciNetCrossRefMATHGoogle Scholar
  70. [L.S]
    T. Lyons and D. Sullivan, Function theory,random paths and covering spaces, J. Diff. Geo, 19, 1984, 299–323.MathSciNetMATHGoogle Scholar
  71. [Marg]
    G. A. Margulis, Positive harmonic functions on nilpotent groups, Doklady Akad,166, 5, 1966,1054–1057,et Sov. Math. 7, 1966, 241–243.MathSciNetMATHGoogle Scholar
  72. [Mar]
    R. S. Martin, Minmal positive harmonic functions, Trans.Amer.Soc., 49, 1941, 137–172.CrossRefGoogle Scholar
  73. [Me]
    P. A. Meyer, Probabilités et Potentiel,Hermann Act. Sci. Indus., 1318, 1966Google Scholar
  74. [Mol]
    S.A. Molchanov, On Martin Boundaries for the direct product of Markov chains, Theor. Prob. and Appl. 12, 1967, 307–310.CrossRefMATHGoogle Scholar
  75. [Na]
    L. Naim, Sur le rôle de la frontière de Martin en theorie du Potentiel, Ann. Inst. Fourier, 7, 1957, 183–281MathSciNetCrossRefMATHGoogle Scholar
  76. [Pi]
    M. A. Pinsky, Stochastic Riemannian Geometry, Probabilistic Methods in Analysis and related topics, Bharucha-Reid Ed., Acad. Press, New-York, 199–236Google Scholar
  77. [Pra]
    J.J. Prat, Etude aymptotique et convergence angulaire du mouvement Brownien sur une variété à courbure négative, C.R.acad. Sci. 290, 1975, 1539–1542.MathSciNetMATHGoogle Scholar
  78. [Rev]
    D. Revuz, Markov chains, North Holland, Amsterdam 1975.MATHGoogle Scholar
  79. [Seg]
    D.Segal, Polycyclic groups,Cambridge tracts in Math., 82, Cambridge University Press, 1983.Google Scholar
  80. [Ser]
    J. Serrin, On the Harnack inequality for linear elliptic equations, J. Anal. Math., 4,1956, 292–308.MathSciNetCrossRefMATHGoogle Scholar
  81. [Sib]
    D. Sibony, Théorème de limites fines et problème de Dirichlet, Ann. Inst. Fourier, XVIII (2) 1968,121–134.MathSciNetCrossRefMATHGoogle Scholar
  82. [Sul.1]
    D. Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, Riemann surfaces and Related Topics, Proc. of the 1978 Stony Brooks Conference.Google Scholar
  83. [Sul.2]
    D. Sullivan, The Dirichlet problem at infinity for a negatively curved manifold, J. Diff. Geo. 18, 1983, 723–732.MathSciNetMATHGoogle Scholar
  84. [Tay]
    J.C. Taylor, Product of minimals are minimals. A paraitre.Google Scholar
  85. [Var.1]
    N. T. Varopoulos, Brownian motion and random walks on manifolds, Ann. Inst. Fourier,34, 1984, 243–269.MathSciNetCrossRefMATHGoogle Scholar
  86. [Var.2]
    N. T. Varopoulos, Random walks on soluble groups, Bull. Sci. Math.107, 1983, 337–344.MathSciNetMATHGoogle Scholar
  87. [Var.3]
    N. T. Varopoulos, Theorie du Potentiel sur des groupes et des variétés, C. R. Acad. Sci. Paris, 302, 1986,203–205.MathSciNetMATHGoogle Scholar
  88. [Var.4]
    N. T. Varopoulos, Information theory and harmonic functions, Bull. Sci. Math. 110,no4, 1986,347–389.MathSciNetMATHGoogle Scholar
  89. [Var.5]
    N. T. Varopoulos, Isoperimetric inequalities and Markov chains, J.Func.Anal., 63, 1985, 215–239MathSciNetCrossRefMATHGoogle Scholar
  90. [Var.6]
    N. T. Varopoulos,Potential theory and diffusion on Riemannian manifolds, Conference in Harmonic Analysis in Honor of Antoni Zygmund, Wadsworth, Belmont, California, 183.Google Scholar
  91. [Wil]
    R. Williams, Diffusions, Markov processes and Martingales, J. Willey, 1979Google Scholar
  92. [Yau]
    S.T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math., XXVIII,1975, 201–228.MathSciNetCrossRefMATHGoogle Scholar

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  • A. Ancona

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