The geometry of Markov diffusion generators
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 9 (2000) no. 2, pp. 305-366. 
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     author = {Ledoux, Michel},
     title = {The geometry of {Markov} diffusion generators},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {305--366},
     publisher = {Universit\'e Paul Sabatier. Facult\'e des sciences},
     address = {Toulouse},
     volume = {Ser. 6, 9},
     number = {2},
     year = {2000},
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     zbl = {0980.60097},
     language = {en},
     url = {http://archive.numdam.org/item/AFST_2000_6_9_2_305_0/}
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Ledoux, Michel. The geometry of Markov diffusion generators. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 9 (2000) no. 2, pp. 305-366. http://archive.numdam.org/item/AFST_2000_6_9_2_305_0/

[Au] Aubin (Th.). - Nonlinear analysis on manifolds. Monge-Ampère equations. Springer (1982). | MR | Zbl

[Ba1] Bakry (D.). - Transformations de Riesz pour les semigroupes symétriques. Séminaire de Probabilités XIX. Lecture Notes in Math. 1123, 2130-174 (1985). Springer. | Numdam | Zbl

[Ba2] Bakry (D.). - Inégalités de Sobolev faibles : un critère Γ2. Séminaire de Probabilités XXV. Lecture Notes in Math. 1485, 234-261 (1991). Springer. | Numdam | MR | Zbl

[Ba3] Bakry (D.). - L'hypercontractivité et son utilisation en théorie des semigroupes. Ecole d'Eté de Probabilités de St-Flour. Lecture Notes in Math. 1581, 1-114 (1994). Springer. | MR | Zbl

[Ba4] Bakry (D.). - On Sobolev and logarithmic Sobolev inequalities for Markov semigroups. New trends in Stochastic Analysis. 43-75 (1997). World Scientific | MR

[Ba-E] Bakry (D.), Emery (M.). - Diffusions hypercontractives. Séminaire de Probabilités XIX. Lecture Notes in Math. 1123, 177-206 (1985). Springer. | Numdam | MR | Zbl

[B-C-L] Bakry (D.), Concordet (D.), Ledoux (M.). - Optimal heat kernel bounds under logarithmic Sobolev inequalities. ESAIM: Probability and Statistics 1, 391-407 (1997). | Numdam | MR | Zbl

[B-C-L-SC] Bakry (D.), Coulhon (T.), Ledoux (M.), Saloff-Coste (L.). - Sobolev inequalities in disguise. Indiana J. Math. 44, 1033-1074 (1995). | MR | Zbl

[B-L1] Bakry (D.), Ledoux (M.). - Sobolev inequalities and Myers's diameter theorem for an abstract Markov generator. Duke Math. J. 85, 253-270 (1996). | MR | Zbl

[B-L2] Bakry (D.), Ledoux (M.). - Lévy-Gromov's isoperimetric inequality for an infinite dimensional diffusion generator. Invent. math. 123, 259-281 (1996). | MR | Zbl

[B-L-Q] Bakry (D.), Ledoux (M.), Qian (Z.). - Preprint (1997).

[B-Q] Bakry (D.), Qian (Z.). - Comparison theorem for spectral gap via dimension, diameter and Ricci curvature. Preprint (1998).

[Be1] Beckner (W.). - Sobolev inequalities, the Poisson semigroup and analysis on the sphere Sn. Proc. Nat. Acad. Sci. 89, 4816-4819 (1992). | MR | Zbl

[Be2] Beckner (W.). - Personal communication (1998).

[Bé] Bérard (P.H.). - Spectral geometry: Direct and inverse problems. Lecture Notes in Math. 1207 (1986). Springer. | MR | Zbl

[BV-V] Bidaut-Veron (M.-F.), Veron (L.). - Nonlinear elliptic equations on compact manifolds and asymptotics of Emden equations. Invent. math. 106, 489-539 (1991). | MR | Zbl

[Bob] Bobkov (S.). - An isoperimetric inequality on the discrete cube and an elementary proof of the isoperimetric inequality in Gauss space. Ann. Probability 25, 206-214 (1997). | MR | Zbl

[B-H] Bobkov (S.), Houdré (Ch.). - Some connections between Sobolev-type inequalities and isoperimetry. Memoirs of the A.M.S. 616 (1997). | MR | Zbl

[Bor] Borell (C.). - The Brunn-Minkowski inequality in Gauss space. Invent. math. 30, 207-216 (1975). | MR | Zbl

[B-Z] Burago (Y.D.), Zalgaller (V.A.). - Geometric inequalities. Springer (1988). First Edition (russian): Nauka (1980). | MR | Zbl

[Ca] Carlen (E.). - Superadditivity of Fisher's information and logarithmic Sobolev inequalities. J. Funct. Anal. 101, 194-211 (1991). | MR | Zbl

[C-L] Carlen (E.), Loss (M.). - Sharp constant in Nash's inequality. Duke Math. J., International Math. Research Notices 7, 213-215 (1993). | MR | Zbl

[C-K-S] Carlen (E.), Kusuoka (S.), Stroock (D.). - Upperbounds for symmetric Markov transition functions. Ann. Inst. H. Poincaré 23, 245-287 (1987). | Numdam | MR | Zbl

[Cha1] Chavel (I.). - Eigenvalues in Riemannian geometry. Academic Press (1984). | MR | Zbl

[Cha2] Chavel (I.). - Riemannian geometry - A modern introduction. Cambridge Univ. Press (1993). | MR | Zbl

[Ch] Cheeger (J.). - The relation between the Laplacian and the diameter for manifolds of non-negative curvature. Arch. der Math. 19, 558-560 (1968). | MR | Zbl

[Che] Cheng (S.-Y.). - Eigenvalue comparison theorems and its geometric applications. Math. Z. 143, 289-297 (1975). | MR | Zbl

[Da] Davies (E.B.). - Heat kernel and spectral theory. Cambridge Univ. Press (1989). | MR | Zbl

[D-S] Deuschel (J.-D.), Stroock (D.). - Large deviations. Academic Press (1989). | MR | Zbl

[Dr] Druet (O.). - Optimal Sobolev inequalities of arbitrary order on compact Riemannian manifolds (1998). J. Funct. Anal. 159, 217-242 (1998). | MR | Zbl

[D-H-V] Druet (O.), Hebey (E.), Vaugon (M.). - Optimal Nash's inequalities on Riemannian manifolds: the influence of geometry. International Math. Research Notices 14, 735-779 (1999). | MR | Zbl

[Eh] Ehrhard (A.). - Symétrisation dans l'espace de Gauss. Math. Scand. 53, 281-301 (1983). | MR | Zbl

[F-L-M] Figiel (T.), Lindenstrauss (J.), Milman (V.D.). - The dimensions of almost spherical sections of convex bodies. Acta Math. 139, 52-94 (1977). | MR | Zbl

[Fo] Fontenas (E.). - Sur les constantes de Sobolev des variétés riemanniennes compactes et les fonctions extrémales des sphères. Bull. Sci. math. 121, 71-96 (1997). | MR | Zbl

[G-H-L] Gallot (S.), Hulin (D.), Lafontaine (J.). - Riemannian Geometry. Second Edition. Springer (1990). | MR | Zbl

[Gro] Gromov (M.). - Paul Lévy's isoperimetric inequality. Preprint I.H.E.S. (1980).

[G-M] Gromov (M.), Milman (V.D.). - A topological application of the isoperimetric inequality. Amer. J. Math. 105, 843-854 (1983). | MR | Zbl

[Gr1] Gross (L.). - Logarithmic Sobolev inequalities. Amer. J. Math. 97, 1061-1083 (1975). | MR | Zbl

[Gr2] Gross (L.). - Logarithmic Sobolev inequalities and contractive properties of semigroups. Dirichlet Forms, Varenna 1992. Lect. Notes in Math. 1563, 54-88 (1993). Springer. | MR | Zbl

[He1] Hebey (E.). - Sobolev spaces on Riemannian manifolds. Lecture Notes in Math. 1635. Springer (1996). | MR | Zbl

[He2] Hebey (E.). - Nonlinear analysis on manifolds: Sobolev spaces and inequalities. CIMS Lecture Notes (1999). Courant Institute of Mathematical Sciences. | MR | Zbl

[Il] Ilias (S.). - Constantes explicites pour les inégalités de Sobolev sur les variétés riemanniennnes compactes. Ann. Inst. Fourier 33, Fasc. 2, 151-165 (1983). | Numdam | MR | Zbl

[Kr] Kröger (P.). - On the spectral gap for compact manifolds. J. Differential Geometry 36, 315-330 (1992). | MR | Zbl

[L-O] Latala (R.), Oleszkiewicz (K.). - Between Sobolev and Poincaré (1999). Geometric and Funct. Anal., to appear. | MR | Zbl

[Le1] Ledoux (M.). - L'algèbre de Lie des gradients itérés d'un générateur markovien - Développements de moyennes et entropies. Ann. scient. Éc. Norm. Sup. 28, 435-460 (1995). | Numdam | MR | Zbl

[Le2] Ledoux (M.). - Isoperimetry and Gaussian Analysis. Ecole d'Eté de Probabilités de St-Flour 1994. Lecture Notes in Math. 1648, 165-294 (1996). Springer. | MR | Zbl

[Le3] Ledoux (M.). - On manifolds with non-negative Ricci curvature and Sobolev inequalities. Comm. in Analysis and Geometry 7, 347-353 (1999). | MR | Zbl

[Le4] Ledoux (M.). - Concentration of measure and logarithmic Sobolev inequalities. Séminaire de Probabilités XXXIII. Lecture Notes in Math. 1709, 120-216 (1999). Springer. | Numdam | MR | Zbl

[Lé] Lévy (P.). - Problèmes concrets d'analyse fonctionnelle. Gauthier-Villars (1951). | MR | Zbl

[Li1] Li (P.). - A lower bound for the first eigenvalue of the Laplacian on a compact manifold. Indiana Univ. Math. J. 28, 1013-1019 (1979). | MR | Zbl

[Li2] Li (P.). - Large time behavior of the heat equation on complete manifolds with non-negative Ricci curvature. Ann. Math. 124, 1-21 (1986). | MR | Zbl

[L-Y] Li (P.), Yau (S.-T.). - On the parabolic kernel of the Schrödinger operator. Acta Math. 156, 153-201 (1986). | MR | Zbl

[Lie] Lieb (E.). - Gaussian kernels have only Gaussian maximizers. Invent. math. 102, 179-208 (1990). | MR | Zbl

[MK] Mckean (H.P.). - Geometry of differential space. Ann. Probability 1, 197-206 (1973). | MR | Zbl

[Ma] Mazet (O.). - Classification des semigroupes de diffusion sur Rn associés à une famille de polynômes orthogonaux. Séminaire de Probabilités XXXI. Lecture Notes in Math. 1655, 40-53 (1997). Springer. | Numdam | MR | Zbl

[Mo] Moser (J.). - A Harnack inequality for parabolic differential equations. Comm. Pure Appl. Math. 17, 101-134 (1964). | MR | Zbl

[M-W] Muller (C.), Weissler (F.). - Hypercontractivity of the heat semigroup for ultraspherical polynomials and on the n-sphere. J. Funct. Anal. 48, 252-283 (1982). | MR | Zbl

[My] Myers (S.B.). - Connections between differential geometry and topology. Duke Math. J. 1, 376-391 (1935). | JFM | Zbl

[Na] Nash (J.). - Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80, 931-954 (1958). | MR | Zbl

[Ob] Obata (M.). - Certain conditions for a Riemannian manifold to be isometric with a sphere. J. Math. Soc. Japan 14, 333-340 (1962). | MR | Zbl

[Os] Osserman (R.). - The isoperimetric inequality. Bull. Amer. Math. Soc. 84, 1182-1238 (1978). | MR | Zbl

[Ro1] Rothaus (O.). - Diffusion on compact Riemannian manifolds and logarithmic Sobolev inequalities. J. Funct. Anal. 42, 358-367 (1981). | MR | Zbl

[Ro2] Rothaus (O.). - Hypercontractivity and the Bakry-Emery criterion for compact Lie groups. J. Funct. Anal. 65, 358-367 (1986). | MR | Zbl

[SC] Saloff-Coste (L.). - Convergence to equilibrium and logarithmic Sobolev constant on manifolds with Ricci curvature bounded below. Colloquium Math. 67, 109-121 (1994). | MR | Zbl

[Sc] Schmidt (E.). - Die Brunn-Minkowskische Ungleichung und ihr Spiegelbild sowie die isoperime- trische Eigenschaft der Kugel in der euklidischen und nichteuklidischen Geometrie. Math. Nach. 1, 81-157 (1948). | MR | Zbl

[So] Sobolev (S.L.). - On a theorem in functional analysis. Amer. Math. Soc. Translations (2) 34, 39-68 (1963); translated from Mat. Sb. (N.S.) 4 (46), 471-497 (1938). | MR | Zbl

[S-T] Sudakov (V.N.), Tsirel'Son (B.S.). - Extremal properties of half-spaces for spherically invariant measures. J. Soviet. Math. 9, 9-18 (1978); translated from Zap. Nauch. Sem. L.O.M.I. 41, 14-24 (1974). | MR | Zbl

[Ta] Talenti (G.). - Best constants in Sobolev inequality. Ann. di Matem. Pura ed Appl. 110, 353-372 (1976). | MR | Zbl

[To] Topogonov (V.A.). - Riemannian spaces having their curvature bounded below by a positive number. Usphei Math. Nauk. 14, 87-135 (1959). Transl. Amer. Math. Soc. 37, 291-336 (1964). | Zbl

[Va1] Varopoulos (N.). - Une généralisation du théorème de Hardy-Littlewood-Sobolev pour les espaces de Dirichlet. C. R. Acad. Sci. Paris 299, 651-654 (1984). | MR | Zbl

[Va2] Varopoulos (N.). - Hardy-Littlewood theory for semigroups. J. Funct. Anal. 63, 240-260 (1985). | MR | Zbl

[Va3] Varopoulos (N.). - Analysis and geometry on groups. Proceedings of the International Congress of Mathematicians, Kyoto (1990), vol. II, 951-957 (1991). Springer-Verlag. | MR | Zbl

[Y-Z] Yang (H.C.), Zhong (J.Q.). - On the estimate of the first eigenvalue of a compact Riemannian manifold. Sci. Sinica Ser.A 27, 1265-1273 (1984). | MR | Zbl

[Wa] Wang (F.-Y.). - Logarithmic Sobolev inequalities on noncompact Riemannian manifolds. Probab. Theory Relat. Fields 109, 417-424 (1997). | MR | Zbl

[Yo] Yoshida (K.). - Functional Analysis. Sixth Edition. Springer (1995).