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Theorie du Potentiel sur les Graphes et les Varietes

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École d'Été de Probabilités de Saint-Flour XVIII - 1988

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Références Bibliographiques

  1. H. Aikawa, On the thinness in a Lipschitz domain, Analysis, 5,1985, 345–382.

    Article  MathSciNet  MATH  Google Scholar 

  2. G. Alexopoulos, Fonctions harmoniques bornées sur les groupes résolubles, C.R.Acad.Sci. Paris, 305, 1987, 777–779

    MathSciNet  MATH  Google Scholar 

  3. 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, 169–213

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Ancona, Une propriété de la compactification de Martin d'un domaine euclidien, Ann. Inst. Fourier, XIX, 4, 1979, 71–90.

    Article  MathSciNet  MATH  Google Scholar 

  5. A. Ancona, Negatively curved manifolds,elliptic operators and the Martin Boundary, Annals of Maths, 125, 1987, 495–536.

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Ancona, Positive harmonic functions and hyperbolicity, Potential Theory, Prague 1987, Springer Lecture notes no1344, 1–23.

    Google Scholar 

  7. A. Ancona, Régularité d'accès des bouts et frontière de Martin des domaines euclidiens, J. Math. Pures et Àp., 63, 1984,215–260.

    MathSciNet  MATH  Google Scholar 

  8. M.T. Anderson, The Dirichlet problem at infinity for manifolds of negative curvature, J. Diff. Geo. 18, 1983, 701–721.

    MathSciNet  MATH  Google Scholar 

  9. M.T. Anderson and R. Schoen, Positive harmonic functions on complete manifolds of negative curvature, Ann. of Maths 121, 1985, 429–461.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Avez, Harmonic functions on groups, Differential Geometry and relativity, Reidel, 1976, 27–32.

    Google Scholar 

  11. R. Azencott, Behavior of diffusion semi-groups at infinity, Bull. Soc. Math. France, 102, 1974, 193–240.

    MathSciNet  MATH  Google Scholar 

  12. P. Baldi, N. Lohoué, J. Peyrière, Sur la classification des groupes récurrents, C.R.A.S. Paris, t. 285, 1103–1104, 1977.

    MATH  Google Scholar 

  13. H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, Proc. Lond. Math. Soc., 25, 1972, 603–614.

    Article  MathSciNet  MATH  Google Scholar 

  14. H. Bauer, Harmonishe raume und ihre Potential Theorie, Lecture Notes 22, 1966. Springer.

    Google Scholar 

  15. M. Benedicks, Positive harmonic functions vanishing on the boundary of certain domains of Rn, Ark. för Math. 18, 1, 1980,53–72.

    Article  MathSciNet  MATH  Google Scholar 

  16. A. Beurling, J. Deny, Dirichlet spaces, Proc. Nat. Acad. Sc., 45, 208–215, 1959.

    Article  MathSciNet  MATH  Google Scholar 

  17. R. M. Blumenthal and R. K. Getoor, Markov processes and Potential theory, 1968, Academic Press, New-York and London.

    MATH  Google Scholar 

  18. M. Brelot, Axiomatique des fonctions harmoniques, Les Presses de l'Université de Montréal, 1969.

    Google Scholar 

  19. M. Brelot, Sur le principe des singularités positives et la topologie de R.S.Martin, Ann. Univ. Grenoble, XIII,23, 1947–48,113–142.

    MathSciNet  MATH  Google Scholar 

  20. M. Brelot et J.L. Doob, Limites angulaires et limites fines, Ann. Inst. Fourier 13, 2, 1963, 395–415.

    Article  MathSciNet  MATH  Google Scholar 

  21. R. Brooks, The fundamental group and the spectrum of the laplacian, Comm. Math. Helv.56, 1981, 581–598.

    Article  MathSciNet  MATH  Google Scholar 

  22. I. Chavel, Eigenvalues in Riemannian geometry, Academic Press Inc., 1984

    Google Scholar 

  23. J. Cheeger and D. Ebin, Comparison theorems in Differential geometry, North Holland Publ. Co, Amsterdam, 1975

    MATH  Google Scholar 

  24. 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.

    MathSciNet  MATH  Google Scholar 

  25. S.Y. Cheng and S.T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math., XXVIII, 1975, 333–354.

    Article  MathSciNet  MATH  Google Scholar 

  26. 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.

    Article  MathSciNet  MATH  Google Scholar 

  27. G. Choquet, Lectures on Analysis, tome 2, W. A. Benjamin Inc, 1969.

    Google Scholar 

  28. G. Choquet, J. Deny, Sur l'équation de convolution μ=μ*σ, C.R. Acad. Sci. 250,1960,799–801.

    MathSciNet  MATH  Google Scholar 

  29. C.Constantinescu, A.Cornea, Potential theory on harmonic spaces, 1972, Springer Verlag.

    Google Scholar 

  30. E. B. Davies, Heat kernels and spectral theory, Cambridge tracts in Mathematics, no92, 1989.

    Google Scholar 

  31. J.Deny, Méthodes hilbertiennes en théorie du Potentiel, Cours de Stresa, C.I.M.E., Jul. 1969.

    Google Scholar 

  32. Y. Derriennic, Lois "zéro ou deux" pour les processus de Markov. Applications aux marches aléatoires, Ann. Inst. Poincaré, XII,2, 1976, 111–129.

    MathSciNet  MATH  Google Scholar 

  33. Y. Derriennic, Quelques applications du théorème ergodique sous-additif. Astérisque, 74, 183–201.

    Google Scholar 

  34. J. L. Doob, Classical Potential theory and its probabilistic counterpart, Springer Verlag, New-York, 1984.

    Book  MATH  Google Scholar 

  35. J. L. Doob, Conditionned Brownian motion and the boundary limits of harmonic functions, Bull. Soc. Math. de France, 85, 1957,431–458.

    MathSciNet  MATH  Google Scholar 

  36. E. Dynkin, Markov Processes, Springer Verlag, Vol. 1–2,New-York, 1965.

    Google Scholar 

  37. P. Eberlein and B. O'Neill, Visibility manifolds, Pac. J. of Math. 46, 1973, 45–109.

    Article  MathSciNet  MATH  Google Scholar 

  38. 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. J.L. Fernandez, On the existence of Green's function in Riemannian manifolds, Proc of the A.M.S., 96,2, 1986, 284–286.

    Article  MATH  Google Scholar 

  40. M. Freire, Positive harmonic functions on product of manifolds, Preprint.

    Google Scholar 

  41. A. Friedman, Partial Differential Equations of Parabolic type, Englewood Cliffs, NJ, Prentice Hall, 1964.

    MATH  Google Scholar 

  42. M.Fukushima, Dirichlet forms ans Markov processes, North-Holland, 1980.

    Google Scholar 

  43. H. Furstenberg, A Poisson formula for semi-simple Lie Groups, Ann. of Math.77,2, 1963, 335–386.

    Article  MathSciNet  MATH  Google Scholar 

  44. D. Gildbarg and N. S. Trudinger, Elliptic partial differential equations of the second order,2nd edition, Springer, New-York,1983.

    Book  Google Scholar 

  45. K. Gowrisankaran, Fatou-Doob limit theorems in the axiomatic setting of Brelot, Ann. Inst. Fourrier,XVI, 2, 1966, 465–467.

    MathSciNet  MATH  Google Scholar 

  46. F.P. Greenleaf, Invariant means on topological groups, Van Nostrand, 1969.

    Google Scholar 

  47. M. Gromov, Hyperbolic groups, Essays in group theory, M.S.R.I Publications, 8, 1987, 75–263.

    Article  MathSciNet  Google Scholar 

  48. M. Gromov, Groups of polynomial growth and expanding maps, Publications de l'I.H.E.S, 53,1981, 53–78.

    MathSciNet  MATH  Google Scholar 

  49. Y. Guivarc'h, Mouvement brownien sur les revêtements d'une variété compacte, C.R.Acad. Sc. Paris,892, 1981,851–853.

    MathSciNet  MATH  Google Scholar 

  50. 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. 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. Y.Guivarc'h, M.Keane, B.Roynette, Marches aléatoires sur les groupes de Lie, Lecture Notes in Mathematics, 624, Springer Verlag.

    Google Scholar 

  53. R. M. Hervé, Recherches sur la théorie des fonctions surharmoniques et du potentiel, Ann. Inst. Fourier, XII, 1962, 415–471.

    Article  MATH  Google Scholar 

  54. R.A. Hunt, R.L. Wheeden, Positive Harmonic functions on Lipschitz doamins, Trans. Amer. Math. Soc., 147,1970, 507–527.

    Article  MathSciNet  MATH  Google Scholar 

  55. D. Jerison, C. Kenig, Boundary behavior of harmonic functions in non-tangentially accessible domains, Advances in Math.,46, 1982, 80–147.

    Article  MathSciNet  MATH  Google Scholar 

  56. 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. V. A. Kaimanovich and A. M. Vershik, Random walks on discrete groups: Boundary and entropy, Ann. Proba. 11, 3, 1983, 457–490.

    Article  MathSciNet  MATH  Google Scholar 

  58. L. Karp and P. Li, The heat equation on complete Riemannain manifolds. Pretirage.

    Google Scholar 

  59. 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.

    MathSciNet  MATH  Google Scholar 

  60. Y.Kifer, F.Ledrappier, Hausdorff dimension of harmonic measures on negatively curved manifolds, à paraitre in Trans. Amer. Math. Soc.

    Google Scholar 

  61. 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.

    Article  MathSciNet  MATH  Google Scholar 

  62. H.Kunita, T.Wanatabe, Markov processes and Martin boundaries, Ill. J. Math., 9, 1965.

    Google Scholar 

  63. F. Ledrappier, Propriété de Poisson et courbure négative, C.R.Acad.Sci. Paris, 305, 1987, 191–194.

    MathSciNet  MATH  Google Scholar 

  64. F. Ledrappier, Ergodic properties of brownian motion on covers of compact negatively curved manifolds, Bol. Soc. Bras. Mat.,19,1988, 115–140.

    Article  MathSciNet  MATH  Google Scholar 

  65. 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.

    MathSciNet  MATH  Google Scholar 

  66. P. Li and S. T. Yau, On the parabolic kernel of the Schrodinger operator, Acta Math., 156, 1986, 153–201.

    Article  MathSciNet  Google Scholar 

  67. T.J. Lyons,Instability of the Liouville property for quasi-isometric Riemannian manifolds and reversible Markov chains, J. Diff. Geo. 26, 1987, 33–66.

    MathSciNet  MATH  Google Scholar 

  68. T.J. Lyons, A simple criterion for transience of a reversible markov chain, Ann. Proba. 11, 1983, 393–402.

    Article  MathSciNet  MATH  Google Scholar 

  69. T. Lyons and H. P. McKean, Winding of Plane brownian motion, Advances in Math., 51, 1984, 212, 225.

    Article  MathSciNet  MATH  Google Scholar 

  70. T. Lyons and D. Sullivan, Function theory,random paths and covering spaces, J. Diff. Geo, 19, 1984, 299–323.

    MathSciNet  MATH  Google Scholar 

  71. G. A. Margulis, Positive harmonic functions on nilpotent groups, Doklady Akad,166, 5, 1966,1054–1057,et Sov. Math. 7, 1966, 241–243.

    MathSciNet  MATH  Google Scholar 

  72. R. S. Martin, Minmal positive harmonic functions, Trans.Amer.Soc., 49, 1941, 137–172.

    Article  Google Scholar 

  73. P. A. Meyer, Probabilités et Potentiel,Hermann Act. Sci. Indus., 1318, 1966

    Google Scholar 

  74. S.A. Molchanov, On Martin Boundaries for the direct product of Markov chains, Theor. Prob. and Appl. 12, 1967, 307–310.

    Article  MATH  Google Scholar 

  75. L. Naim, Sur le rôle de la frontière de Martin en theorie du Potentiel, Ann. Inst. Fourier, 7, 1957, 183–281

    Article  MathSciNet  MATH  Google Scholar 

  76. M. A. Pinsky, Stochastic Riemannian Geometry, Probabilistic Methods in Analysis and related topics, Bharucha-Reid Ed., Acad. Press, New-York, 199–236

    Google Scholar 

  77. 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.

    MathSciNet  MATH  Google Scholar 

  78. D. Revuz, Markov chains, North Holland, Amsterdam 1975.

    MATH  Google Scholar 

  79. D.Segal, Polycyclic groups,Cambridge tracts in Math., 82, Cambridge University Press, 1983.

    Google Scholar 

  80. J. Serrin, On the Harnack inequality for linear elliptic equations, J. Anal. Math., 4,1956, 292–308.

    Article  MathSciNet  MATH  Google Scholar 

  81. D. Sibony, Théorème de limites fines et problème de Dirichlet, Ann. Inst. Fourier, XVIII (2) 1968,121–134.

    Article  MathSciNet  MATH  Google Scholar 

  82. 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. D. Sullivan, The Dirichlet problem at infinity for a negatively curved manifold, J. Diff. Geo. 18, 1983, 723–732.

    MathSciNet  MATH  Google Scholar 

  84. J.C. Taylor, Product of minimals are minimals. A paraitre.

    Google Scholar 

  85. N. T. Varopoulos, Brownian motion and random walks on manifolds, Ann. Inst. Fourier,34, 1984, 243–269.

    Article  MathSciNet  MATH  Google Scholar 

  86. N. T. Varopoulos, Random walks on soluble groups, Bull. Sci. Math.107, 1983, 337–344.

    MathSciNet  MATH  Google Scholar 

  87. N. T. Varopoulos, Theorie du Potentiel sur des groupes et des variétés, C. R. Acad. Sci. Paris, 302, 1986,203–205.

    MathSciNet  MATH  Google Scholar 

  88. N. T. Varopoulos, Information theory and harmonic functions, Bull. Sci. Math. 110,no4, 1986,347–389.

    MathSciNet  MATH  Google Scholar 

  89. N. T. Varopoulos, Isoperimetric inequalities and Markov chains, J.Func.Anal., 63, 1985, 215–239

    Article  MathSciNet  MATH  Google Scholar 

  90. 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. R. Williams, Diffusions, Markov processes and Martingales, J. Willey, 1979

    Google Scholar 

  92. S.T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math., XXVIII,1975, 201–228.

    Article  MathSciNet  MATH  Google Scholar 

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  1. A. Ancona
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Paul-Louis Hennequin

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Ancona, A. (1990). Theorie du Potentiel sur les Graphes et les Varietes. In: Hennequin, PL. (eds) École d'Été de Probabilités de Saint-Flour XVIII - 1988. Lecture Notes in Mathematics, vol 1427. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0103041

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