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Isoperimetry and Gaussian analysis

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References

  • [A-C] R. A. Adams, F. H. Clarke. Gross’s logarithmic Sobolev inequality: a simple proof. Amer. J. Math. 101, 1265–1269 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  • [A-K-S] S. Aida, S. Kusuoka, D. Stroock. On the support of Wiener functionals. Asymptotic problems in probability theory: Wiener functionals and asymptotics. Pitman Research Notes in Math. Series 284, 1–34 (1993). Longman.

    MathSciNet  MATH  Google Scholar 

  • [A-M-S] S. Aida, T. Masuda, I. Shigekawa. Logarithmic Sobolev inequalities and exponential integrability. J. Funct. Anal. 126, 83–101 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  • [A-L-R] M. Aizenman, J. L. Lebowitz, D. Ruelle. Some rigorous results on the Sherrington-Kirkpatrick spin glass model. Comm. Math. Phys. 112, 3–20 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  • [An] T. W. Anderson. The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities. Proc. Amer. Math. Soc. 6, 170–176 (1955).

    Article  MathSciNet  MATH  Google Scholar 

  • [A-G] M. Arcones, E. Giné. On decoupling, series expansions and tail behavior of chaos processes. J. Theoretical Prob. 6, 101–122 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  • [Az] R. Azencott. Grandes déviations et applications. École d’été de Probabilités de St-Flour 1978. Lecture Notes in Math. 774, 1–176 (1978). Springer-Verlag.

    Article  MathSciNet  Google Scholar 

  • [Azu] K. Azuma. Weighted sums of certain dependent random variables. Tohoku Math. J. 19, 357–367 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  • [B-C] A. Badrikian, S. Chevet. Mesures cylindriques, espaces de Wiener et fonctions aléatoires gaussiennes. Lecture Notes in Math 379, (1974). Springer-Verlag.

    Google Scholar 

  • [Ba] A. Baernstein II. Integral means, univalent functions and circular symmetrization. Acta Math. 133, 139–169 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  • [B-T] A. Baernstein II, B. A. Taylor. Spherical rearrangements, subharmonic functions and *-functions in n-space. Duke Math. J. 43, 245–268 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  • [Bak] D. Bakry. L’hypercontractivité et son utilisation en théorie des semigroupes. École d’Été de Probabilités de St-Flour. Lecture Notes in Math. 1581, 1–114 (1994). Springer-Verlag.

    Article  MathSciNet  Google Scholar 

  • [B-É] D. Bakry, M. Émery. Diffusions hypercontractives. Séminaire de Probabilités XIX. Lecture Notes in Math. 1123, 175–206 (1985). Springer-Verlag.

    Article  MathSciNet  MATH  Google Scholar 

  • [B-R] P. Baldi, B. Roynette. Some exact equivalents for Brownian motion in Hölder norm. Prob. Th. Rel. Fields 93, 457–484 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  • [B-BA-K] P. Baldi, G. Ben Arous, G. Kerkyacharian. Large deviations and the Strassen theorem in Hölder norm. Stochastic Processes and Appl. 42, 171–180 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  • [Bas] R. Bass. Probability estimates for multiparameter Brownian processes. Ann. Probability 16, 251–264 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  • [Be1] W. Beckner. Inequalities in Fourier analysis. Ann. Math. 102, 159–182 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  • [Be2] W. Beckner. Unpublished (1982).

    Google Scholar 

  • [Be3] W. Beckner. Sobolev inequalities, the Poisson semigroup and analysis on the sphere S n. Proc. Nat. Acad. Sci. 89, 4816–4819 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  • [Bel] D. R. Bell. The Malliavin calculus. Pitman Monographs 34. Longman (1987).

    Google Scholar 

  • [BA-L1] G. Ben Arous, M. Ledoux. Schilder’s large deviation principle without topology. Asymptotic problems in probability theory: Wiener functionals and asymptotics. Pitman Research Notes in Math. Series 284, 107–121 (1993). Longman.

    Google Scholar 

  • [BA-L2] G. Ben Arous, M. Ledoux. Grandes déviations de Freidlin-Wentzell en norme hölderienne. Séminaire de Probabilités XXVIII. Lecture Notes in Math. 1583, 293–299 (194). Springer-Verlag.

    Google Scholar 

  • [BA-G-L] G. Ben Arous, M.Gradinaru, M. Ledoux. Hölder norms and the support theorem for diffusions. Ann. Inst. H. Poincaré 30, 415–436 (1994).

    MathSciNet  MATH  Google Scholar 

  • [Bob1] S. Bobkov. A functional form of the isoperimetric inequality for the Gaussian measure (1993). To appear in J. Funct. Anal.

    Google Scholar 

  • [Bob2] S. Bobkov. An isoperimetric inequality on the discrete cube and an elementary proof of the isoperimetric inequality in Gauss space. Preprint (1994).

    Google Scholar 

  • [Bog] V. I. Bogachev. Gaussian measures on linear spaces (1994). To appear.

    Google Scholar 

  • [Bon] A. Bonami. Etude des coefficients de Fourier des fonctions de L p (G). Ann. Inst. Fourier 20, 335–402 (1970).

    Article  MathSciNet  MATH  Google Scholar 

  • [Bo1] C. Borell. Convex measures on locally convex spaces. Ark. Mat. 12, 239–252 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  • [Bo2] C. Borell. The Brunn-Minskowski inequality in Gauss space. Invent. Math. 30, 207–216 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  • [Bo3] C. Borell. Gaussian Radon measures on locally convex spaces. Math. Scand. 38, 265–284 (1976).

    MathSciNet  MATH  Google Scholar 

  • [Bo4] C. Borell. A note on Gauss measures which agree on small balls. Ann. Inst. H. Poincaré 13, 231–238 (1977).

    MathSciNet  MATH  Google Scholar 

  • [Bo5] C. Borell. Tail probabilities in Gauss space. Vector Space Measures and Applications, Dublin 1977. Lecture Notes in Math. 644, 71–82 (1978). Springer-Verlag.

    Google Scholar 

  • [Bo6] C. Borell. On the integrability of Banach space valued Walsh polynomials. Séminaire de Probabilités XIII. Lecture Notes in Math. 721, 1–3 (1979). Springer-Verlag.

    Article  MathSciNet  Google Scholar 

  • [Bo7] C. Borell. A Gaussian correlation inequality for certain bodies in ℝn. Math. Ann. 256, 569–573 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  • [Bo8] C. Borell. On polynomials chaos and integrability. Prob. Math. Statist. 3, 191–203 (1984).

    MathSciNet  MATH  Google Scholar 

  • [Bo9] C. Borell. On the Taylor series of a Wiener polynomial. Seminar Notes on multiple stochastic integration, polynomial chaos and their integration. Case Western Reserve University, Cleveland (1984).

    MATH  Google Scholar 

  • [Bo10] C. Borell. Geometric bounds on the Ornstein-Uhlenbeck process. Z. Wahrscheinlichkeitstheor. verw. Gebiete 70, 1–13 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  • [Bo11] C. Borell. Intrinsic bounds on some real-valued stationary random functions. Probability in Banach spaces V. Lecture Notes in Math. 1153, 72–95 (1985). Springer-Verlag.

    Article  MathSciNet  Google Scholar 

  • [Bo12] C. Borell. Analytic and empirical evidence of isoperimetric processes. Probability in Banach spaces 6. Progress in Probability 20, 13–40 (1990). Birkhäuser.

    MathSciNet  Google Scholar 

  • [B-M] A. Borovkov, A. Mogulskii. On probabilities of small deviations for stochastic processes. Siberian Adv. Math. 1, 39–63 (1991).

    MathSciNet  MATH  Google Scholar 

  • [B-L-L] H. Brascamp, E. H. Lieb, J. M. Luttinger. A general rearrangement inequality for multiple integrals. J. Funct. Anal. 17, 227–237 (1974).

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  • [C-M] R. H. Cameron, W. T. Martin. Transformations of Wiener integrals under translations. Ann. Math. 45, 386–396 (1944).

    Article  MathSciNet  MATH  Google Scholar 

  • [Ca] M. Capitaine. Onsager-Machlup functional for some smooth norms on Wiener space (1994). To appear in Prob. Th. Rel. Fields.

    Google Scholar 

  • [Ca-L] E. Carlen, M. Loss. Extremals of functionals with competing symmetries. J. Funct. Anal. 88, 437–456 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  • [C-F] I. Chavel, E. Feldman. Modified isoperimetric constants, and large time heat diffusion in Riemannian manifold. Duke Math. J. 64, 473–499 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  • [Ch] J. Cheeger. A lower bound for the smallest eigenvalue of the Laplacian. Problems in Analysis, Symposium in honor of S. Bochner, 195–199, Princeton Univ. Press, Princeton (1970).

    MATH  Google Scholar 

  • [C-L-Y] S. Cheng, P. Li, S.-T. Yau. On the upper estimate of the heat kernel on a complete Riemannian manifold. Amer. J. Math. 156, 153–201 (1986).

    MathSciNet  Google Scholar 

  • [Che] S. Chevet. Gaussian measures and large deviations. Probability in Banach spaces IV. Lecture Notes in Math. 990, 30–46 (1983). Springer-Verlag.

    Article  MathSciNet  Google Scholar 

  • [Ci1] Z. Ciesielski. On the isomorphisms of the spaces H α and m. Bull. Acad. Pol. Sc. 8, 217–222 (1960).

    MathSciNet  MATH  Google Scholar 

  • [Ci2] Z. Ciesielski. Orlicz spaces, spline systems and brownian motion. Constr. Approx. 9, 191–208 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  • [Co] F. Comets. A spherical bound for the Sherrington-Kirkpatrick model. Preprint (1994).

    Google Scholar 

  • [C-N] F. Comets, J. Neveu. The Sherrington-Kirkpatrick model of spin glasses and stochastic calculus: the high temperature case. Preprint (1993).

    Google Scholar 

  • [C-L] T. Coulhon, M. Ledoux. Isopérimétrie, décroissance du noyau de la chaleur et transformations de Riesz: un contre-exemple. Ark. Mat. 32, 63–77 (1994).

    Article  MathSciNet  Google Scholar 

  • [DG-E-...] S. Das Gupta, M. L. Eaton, I. Olkin, M. Perlman, L. J. Savage, M. Sobel. Inequalities on the probability content of convex regions for elliptically contoured distributions. Proc. Sixth Berkeley Symp. Math. Statist. Prob. 2, 241–264 (1972). Univ. of California Press.

    MATH  Google Scholar 

  • [Da] E. B. Davies. Heat kernels and spectral theory. Cambridge Univ. Press (1989).

    Google Scholar 

  • [Da-S] E. B. Davies, B. Simon. Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians. J. Funct. Anal. 59, 335–395 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  • [D-L] P. Deheuvels, M. A. Lifshits. Strassen-type functional laws for strong topologies. Prob. Th. Rel. Fields 97, 151–167 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  • [De] J. Delporte. Fonctions aléatoires presque sûrement continues sur un intervalle fermé. Ann. Inst. H. Poincaré 1, 111–215 (1964).

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  • [D-F] P. Diaconis, D. Freedman. A dozen de Finetti-style results in search of a theory. Ann. Inst. H. Poincaré 23, 397–423 (1987).

    MathSciNet  MATH  Google Scholar 

  • [D-V] M. D. Donsker, S. R. S. Varadhan. Asymptotic evaluation of certain Markov process expectations for large time III. Comm. Pure Appl. Math. 29, 389–461 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  • [Du1] R. M. Dudley. The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Funct. Anal. 1, 290–330 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  • [Du2] R. M. Dudley. Sample functions of the Gaussian process. Ann. Probability 1, 66–103 (1973).

    Article  MathSciNet  MATH  Google Scholar 

  • [D-HJ-S] R. M. Dudley, J. Hoffmann-Jorgensen, L. A. Shepp. On the lower tail of Gaussianseminorms. Ann. Probability 7, 319–342 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  • [Dv] A. Dvoretzky. Some results on convex bodies and Banach spaces. Proc. Symp. on Linear Spaces, Jerusalem, 123–160 (1961).

    Google Scholar 

  • [Eh1] A. Ehrhard. Une démonstration de l’inégalité de Borell. Ann. Scientifiques de l’Université de Clermont-Ferrand 69, 165–184 (1981).

    MathSciNet  MATH  Google Scholar 

  • [Eh2] A. Ehrhard. Symétrisation dans l’espace de Gauss. Math. Scand. 53, 281–301 (1983).

    MathSciNet  MATH  Google Scholar 

  • [Eh3] A. Ehrhard. Inégalités isopérimétriques et intégrales de Dirichlet gaussiennes. Ann. scient. Éc. Norm. Sup. 17, 317–332 (1984).

    MathSciNet  MATH  Google Scholar 

  • [Eh4] A. Ehrhard. Sur l’inégalité de Sobolev logarithmique de Gross. Séminaire de Probabilités XVIII. Lecture Notes in Math. 1059, 194–196 (1984). Springer-Verlag.

    Article  MathSciNet  MATH  Google Scholar 

  • [Eh5] A. Ehrhard. Eléments extrémaux pour les inégalités de Brunn-Minkowski gaussiennes. Ann. Inst. H. Poincaré 22, 149–168 (1986).

    MathSciNet  MATH  Google Scholar 

  • [E-S] O. Enchev, D. Stroock. Rademacher’s theorem for Wiener functionals. Ann. Probability 21, 25–33 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  • [Fa] S. Fang. On the Ornstein-Uhlenbeck process. Stochastics and Stochastic Reports 46, 141–159 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  • [Fed] H. Federer. Geometric measure theory. Springer-Verlag (1969).

    Google Scholar 

  • [F-F] H. Federer, W. H. Fleming. Normal and integral current. Ann. Math. 72, 458–520 (1960).

    Article  MathSciNet  MATH  Google Scholar 

  • [Fe1] X. Fernique. Continuité des processus gaussiens. C. R. Acad. Sci. Paris 258, 6058–6060 (1964).

    MathSciNet  MATH  Google Scholar 

  • [Fe2] X. Fernique Intégrabilité des vecteurs gaussiens. C. R. Acad. Sci. Paris 270, 1698–1699 (1970).

    MathSciNet  MATH  Google Scholar 

  • [Fe3] X. Fernique. Régularité des processus gaussiens. Invent. Math. 12, 304–320 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  • [Fe4] X. Fernique. Régularité des trajectoires des fonctions aléatoires gaussiennes. École d’Été de Probabilités de St-Flour 1974. Lecture Notes in Math. 480, 1–96 (1975). Springer-Verlag.

    Article  MathSciNet  Google Scholar 

  • [Fe5] X. Fernique. Gaussian random vectors and their reproducing kernel Hilbert spaces. Technical report, University of Ottawa (1985).

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • [F-W1] M. Freidlin, A. Wentzell. On small random perturbations of dynamical systems. Russian Math. Surveys 25, 1–55 (1970).

    MathSciNet  MATH  Google Scholar 

  • [F-W2] M. Freidlin, A. Wentzell. Random perturbations of dynamical systems. Springer-Verlag (1984).

    Google Scholar 

  • [Ga] E. Gagliardo. Proprieta di alcune classi di funzioni in piu variabili. Ricerche Mat. 7, 102–137 (1958).

    MathSciNet  MATH  Google Scholar 

  • [G-R-R] A. M. Garsia, E. Rodemich, H. Rumsey Jr., A real variable lemma and the continuity of paths of some Gaussian processes. Indiana Math. J. 20, 565–578 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  • [Gal] L. Gallardo. Au sujet du contenu probabiliste d’un lemme d’Henri Poincaré. Ann. Scientifiques de l’Université de Clermont-Ferrand 69, 185–190 (1981).

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  • [Go1] V. Goodman. Characteristics of normal samples. Ann. Probability 16, 1281–1290 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  • [Go2] V. Goodman. Some probability and entropy estimates for Gaussian measures. Probability in Banach spaces 6. Progress in Probability 20, 150–156 (1990). Birkhäuser.

    Google Scholar 

  • [G-K1] V. Goodman, J. Kuelbs. Cramér functional estimates for Gaussian measures. Diffusion processes and related topics in Analysis. Progress in Probability 22, 473–495 (1990). Birkhäuser.

    MathSciNet  MATH  Google Scholar 

  • [G-K2] V. Goodman, J. Kuelbs. Gaussian chaos and functional laws of the iterated logarithm for Ito-Wiener integrals. Ann. Inst. H. Poincaré 29, 485–512 (1993).

    MathSciNet  MATH  Google Scholar 

  • [Gri] K. Grill. Exact convergence rate in Strassen’s law of the iterated logarithm. J. Theoretical Prob. 5, 197–204 (1991).

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • [Gr1] L. Gross. Abstract Wiener spaces. Proc. 5th Berkeley Symp. Math. Stat. Prob. 2, 31–42 (1965).

    Google Scholar 

  • [Gr2] L. Gross. Potential theory on Hilbert space. J. Funct. Anal. 1, 123–181 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  • [Gr3] L. Gross. Logarithmic Sobolev inequalities. Amer. J. Math. 97, 1061–1083 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  • [Gr4] L. Gross. Logarithmic Sobolev inequalities and contractive properties of semigroups. Dirichlet forms, Varenna (Italy) 1992. Lecture Notes in Math. 1563, 54–88 (1993). Springer-Verlag.

    Article  Google Scholar 

  • [G-N-SS] I. Gyöngy, D. Nualart, M. Sanz-Solé. Approximation and support theorems in modulus spaces (1994). To appear in Prob. Th. Rel. Fields.

    Google Scholar 

  • [Ha] H. Hadwiger. Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Springer-Verlag (1957).

    Google Scholar 

  • [Har] L. H. Harper. Optimal numbering and isoperimetric problems on graphs. J. Comb. Th. 1, 385–393 (1966).

    Article  MathSciNet  MATH  Google Scholar 

  • [He] B. Heinkel. Mesures majorantes et régularité de fonctions aléatoires. Aspects Statistiques et Aspects Physiques des Processus Gaussiens, St-Flour 1980. Colloque C.N.R.S. 307, 407–434 (1980).

    MathSciNet  Google Scholar 

  • [I-S-T] I. A. Ibragimov, V. N. Sudakov, B. S. Tsirel’son. Norms of Gaussian sample functions. Proceedings of the third Japan-USSR Symposium on Probability Theory. Lecture Notes in Math. 550, 20–41 (1976). Springer-Verlag.

    Article  MathSciNet  Google Scholar 

  • [I-W] N. Ikeda, S. Watanabe. Stochastic differential equations and diffusion processes. North-Holland (1989).

    Google Scholar 

  • [It] K. Itô. Multiple Wiener integrals. J. Math. Soc. Japan 3, 157–164 (1951).

    Article  MathSciNet  MATH  Google Scholar 

  • [Ka1] J.-P. Kahane. Sur les sommes vectorielles ∑±u n . C. R. Acad. Sci. Paris 259, 2577–2580 (1964).

    MathSciNet  MATH  Google Scholar 

  • [Ka2] J.-P. Kahane. Some random series of functions. Health Math. Monographs (1968). Second Edition: Cambridge Univ. Press (1985).

    Google Scholar 

  • [Ke] H. Kesten. On the speed of convergence in first-passage percolation. Ann. Appl. Probability 3, 296–338 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  • [Kh] C. Khatri. On certain inequalities for normal distributions and their applications to simultaneous confidence bounds. Ann. Math. Statist. 38, 1853–1867 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  • [K-L1] J. Kuelbs, W. Li. Small ball probabilities for Brownian motion and the Brownian sheet. J. Theoretical Prob. 6, 547–577 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  • [K-L2] J. Kuelbs, W. Li. Metric entropy and the small ball problem for Gaussian measures J. Funct. Anal. 116, 133–157 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  • [K-L-L] J. Kuelbs, W. Li, W. Linde. The Gaussian measure of shifted balls. Prob. Th. Rel. Fields 98, 143–162 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  • [K-L-S] J. Kuelbs, W. Li, Q.-M. Shao. Small ball probabilities for Gaussian processes with stationary increments under Hölder norms (1993). To appear in J. Theoretical Prob..

    Google Scholar 

  • [K-L-T] J. Kuelbs, W. Li, M. Talagrand. Liminf results for Gaussian samples and Chung’s functional LIL. Ann. Probability 22, 1879–1903 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  • [Ku] H.-H. Kuo. Gaussian measures in Banach spaces. Lecture Notes in Math. 436 (1975). Springer-Verlag.

    Google Scholar 

  • [Kus] S. Kusuoka. A diffusion process on a fractal. Probabilistic methods in mathematical physics. Proc. of Taniguchi International Symp. 1985, 251–274. Kinokuniga, Tokyo (1987).

    Google Scholar 

  • [Kw] S. Kwapień. A theorem on the Rademacher series with vector valued coefficients. Probability in Banach Spaces, Obserwolfach 1975. Lecture Notes in Math. 526, 157–158 (1976). Springer-Verlag.

    Article  MathSciNet  MATH  Google Scholar 

  • [K-S] S. Kwapień, J. Sawa. On some conjecture concerning Gaussian measures of dilatations of convex symmetric sets. Studia Math. 105, 173–187 (1993).

    MathSciNet  MATH  Google Scholar 

  • [L-S] H. J. Landau, L. A. Shepp. On the supremum of a Gaussian process. Sankhyà A32, 369–378 (1970).

    MathSciNet  MATH  Google Scholar 

  • [La] R. Latala. A note on the Ehrhard inequality. Preprint (1994).

    Google Scholar 

  • [L-O] R. Latala, K. Oleszkiewicz. On the best constant in the Khintchine-Kahane inequality. Studia Math. 109, 101–104 (1994).

    MathSciNet  MATH  Google Scholar 

  • [Led1] M. Ledoux. Isopérimétrie et inégalités de Sobolev logarithmiques gaussiennes. C. R. Acad. Sci. Paris 306, 79–82 (1988).

    MathSciNet  MATH  Google Scholar 

  • [Led2] M. Ledoux. A note on large deviations for Wiener chaos. Séminaire de Probabilités XXIV, Lecture Notes in Math. 1426, 1–14 (1990). Springer-Verlag.

    Article  MathSciNet  MATH  Google Scholar 

  • [Led3] M. Ledoux. On an integral criterion for hypercontractivity of diffusion semigroups and extremal functions. J. Funct. Anal. 105, 444–465 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  • [Led4] M. Ledoux. A heat semigroup approach to concentration on the sphere and on a compact Riemannian manifold. Geom. and Funct. Anal. 2, 221–224 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  • [Led5] M. Ledoux. Semigroup proofs of the isoperimetric inequality in Euclidean and Gauss space. Bull. Sci. math. 118, 485–510 (1994).

    MathSciNet  MATH  Google Scholar 

  • [L-T1] M. Ledoux, M. Talagrand. Characterization of the law of the iterated logarithm in Banach spaces. Ann. Probability 16, 1242–1264 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  • [L-T2] M. Ledoux, M. Talagrand. Probability in Banach spaces (Isoperimetry and processes). Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag (1991).

    Google Scholar 

  • [Lé] P. Lévy. Problèmes concrets d’analyse fonctionnelle. Gauthier-Villars (1951).

    Google Scholar 

  • [Li] W. Li. Comparison results for the lower tail of Gaussian semi-norms. J. Theoretical Prob. 5, 1–31 (1992).

    Article  Google Scholar 

  • [Li-S] W. Li, Q.-M. Shao. Small ball estimates for Gaussian processes under Sobolev type norms. Preprint (1994).

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • [Lif1] M. A. Lifshits. On the distribution of the maximum of a Gaussian process. Probability Theory and its Appl. 31, 125–132 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  • [Lif2] M. A. Lifshits. Tail probabilities of Gaussian suprema and Laplace transform. Ann. Inst. H. Poincaré 30, 163–180 (1994).

    MathSciNet  MATH  Google Scholar 

  • [Lif3] M. A. Lifshits. Gaussian random functions (1994). Kluwer, to appear.

    Google Scholar 

  • [Lif-T] M. A. Lifshits, B. S. Tsirel’son. Small deviations of Gaussian fields. Probability Theory and its Appl. 31, 557–558 (1987).

    Article  MATH  Google Scholar 

  • [L-Z] T. Lyons, W. Zheng. A crossing estimate for the canonical process on a Dirichlet space and tightness result. Colloque Paul Lévy, Astérisque 157–158, 249–272 (1988).

    Google Scholar 

  • [MD] C. J. H. McDiarmid. On the method of bounded differeces. Twelfth British Combinatorial Conference. Surveys in Combinatorics, 148–188 (1989). Cambrige Univ. Press.

    Google Scholar 

  • [MK] H. P. McKean. Geometry of differential space. Ann. Probability 1, 197–206 (1973).

    Article  MathSciNet  MATH  Google Scholar 

  • [M-P] M. B. Marcus, G. Pisier. Random Fourier series with applications to harmonic analysis. Ann. Math. Studies, vol. 101 (1981). Princeton Univ. Press.

    Google Scholar 

  • [M-S] M. B. Marcus, L. A. Shepp. Sample behavior of Gaussian processes. Proc. of the Sixth Berkeley Symposium on Math. Statist. and Prob. 2, 423–441 (1972).

    MathSciNet  MATH  Google Scholar 

  • [Ma1] B. Maurey. Constructions de suites symétriques. C. R. Acad Sci. Paris 288, 679–681 (1979).

    MathSciNet  MATH  Google Scholar 

  • [Ma2] B. Maurey. Sous-espaces ℓp des espaces de Banach. Séminaire Bourbaki, exp. 608. Astérisque 105–106, 199–216 (1983).

    Google Scholar 

  • [Ma3] B. Maurey. Some deviations inequalities. Geometric and Fucnt. Anal. 1, 188–197 (1991).

    MathSciNet  MATH  Google Scholar 

  • [MW-N-PA] E. Mayer-Wolf, D. Nualart, V. Perez-Abreu. Large deviations for multiple Wiener-Itô integrals. Séminaire de Probabilités XXVI. Lecture Notes in Math. 1526, 11–31 (1992). Springer-Verlag.

    Article  MathSciNet  MATH  Google Scholar 

  • [Maz1] V. G. Maz’ja. Classes of domains and imbedding theorems for function spaces. Soviet Math. Dokl. 1, 882–885 (1960).

    MathSciNet  Google Scholar 

  • [Maz2] V. G. Maz’ja, Sobolev spaces. Springer-Verlag (1985).

    Google Scholar 

  • [Me] M. Mellouk. Support des diffusions dans les espaces de Besov-Orlicz. C. R. Acad. Sci. Paris 319, 261–266 (1994).

    MathSciNet  MATH  Google Scholar 

  • [M-SS] A. Millet, M. Sanz-Solé. A simple proof of the support theorem for diffusion processes. Séminaire de Probabilités XXVIII, Lecture Notes in Math. 1583, 36–48 (1994). Springer-Verlag.

    Article  MathSciNet  MATH  Google Scholar 

  • [Mi1] V. D. Milman. New proof of the theorem of Dvoretzky on sections of convex bodies. Funct. Anal. Appl. 5, 28–37 (1971).

    MathSciNet  Google Scholar 

  • [Mi2] V. D. Milman. The heritage of P. Lévy in geometrical functional analysis. Colloque Paul Lévy sur les processus stochastiques. Astérisque 157–158, 273–302 (1988).

    Google Scholar 

  • [Mi3] V. D. Milman. Dvoretzky’s theorem—Thirty years later (Survey). Geometric and Funct. Anal. 2, 455–479 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  • [Mi-S] V. D. Milman, G. Schechtman. Asymptotic theory of finite dimensional normed spaces. Lecture Notes in Math. 1200 (1986). Springer-Verlag.

    Google Scholar 

  • [M-R] D. Monrad, H. Rootzén. Small values of Gaussian processes and functional laws of the iterated logarithm (1993). To appear in Prob. Th. Rel. Fields.

    Google Scholar 

  • [Mo] J. Moser. On Harnack’s theorem for elliptic differential equations. Comm. Pure Appl. Math. 14, 557–591 (1961).

    Article  MATH  Google Scholar 

  • [Na] J. Nash. Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80, 931–954 (1958).

    Article  MathSciNet  MATH  Google Scholar 

  • [Nel] E. Nelson. The free Markov field. J. Funct. Anal. 12, 211–227 (1973).

    Article  MATH  Google Scholar 

  • [Ne1] J. Neveu. Processus aléatoires gaussiens. Presses de l’Université de Montréal (1968).

    Google Scholar 

  • [Ne2] J. Neveu. Martingales à temps discret. Masson (1972).

    Google Scholar 

  • [Ne3] J. Neveu. Sur l’espérance conditionnelle par rapport à un mouvement brownien. Ann. Inst. H. Poincaré 2, 105–109 (1976).

    MathSciNet  MATH  Google Scholar 

  • [Ni] L. Nirenberg. On elliptic partial differential equations. Ann. Sc. Norm. Sup. Pisa 13, 116–162 (1959).

    MathSciNet  MATH  Google Scholar 

  • [Nu] D. Nualart. The Malliavin calculs and related topics (1994). To appear.

    Google Scholar 

  • [Os] R. Osserman. The isoperimetric inequality. Bull. Amer. Math. Soc. 84, 1182–1238 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  • [Pi1] G. Pisier. Probabilistic methods in the geometry of Banach spaces. Probability and Analysis, Varenna (Italy) 1985. Lecture Notes in Math. 1206, 167–241 (1986). Springer-Verlag.

    Article  MathSciNet  MATH  Google Scholar 

  • [Pi2] G. Pisier. Riesz transforms: a simpler analytic proof of P. A. Meyer inequality. Séminaire de Probabilités XXII. Lecture Notes in Math. 1321, 485–501, Springer-Verlag (1988).

    Article  MathSciNet  Google Scholar 

  • [Pi3] G. Pisier. The volume of convex bodies and Banach space geometry. Cambridge Univ. Press (1989).

    Google Scholar 

  • [Pit] L. Pitt. A Gaussian correlation inequality for symmetric convex sets. Ann. Probability 5, 470–474 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  • [Pr1] C. Preston. Banach spaces arising from some integral inequalities. Indiana Math. J. 20, 997–1015 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  • [Pr2] C. Preston. Continuity properties of some Gaussian processes. Ann. Math. Statist. 43, 285–292 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  • [Sc] M. Schilder. Asymptotic formulas for Wiener integrals. Trans. Amer. Math. Soc. 125, 63–85 (1966).

    Article  MathSciNet  MATH  Google Scholar 

  • [Sch] E. Schmidt. 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).

    Article  Google Scholar 

  • [Sco] A. Scott. A note on conservative confidence regions for the mean value of multivariate normal. Ann. Math. Statist. 38, 278–280 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  • [S-Z1] L. A. Shepp, O. Zeitouni. A note on conditional exponential moments and Onsager-Machlup functionals. Ann. Probability 20, 652–654 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  • [S-Z2] L. A. Shepp, O. Zeitouni. Exponential estimates for convex norms and some applications. Barcelona seminar on Stochastic Analysis, St Feliu de Guixols 1991. Progress in Probability 32, 203–215 (1993). Birkhäuser.

    MathSciNet  MATH  Google Scholar 

  • [Sh] Q.-M. Shao. A note on small ball probability of a Gaussian process with stationary increments. J. Theoretical Prob. 6, 595–602 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  • [S-W] Q.-M. Shao, D. Wang. Small ball probabilities of Gaussian fields. Preprint (1994).

    Google Scholar 

  • [Si] Z. Sidak. Rectangular confidence regions for the means of multivariate normal distributions. J. Amer. Statist. Assoc. 62, 626–633 (1967).

    MathSciNet  MATH  Google Scholar 

  • [Sk] A. V. Skorohod. A note on Gaussian measures in a Banach space. Theor. Probability Appl. 15, 519–520 (1970).

    MathSciNet  Google Scholar 

  • [Sl] D. Slepian. The one-sided barrier problem for Gaussian noise. Bell. System Tech. J. 41, 463–501 (1962).

    Article  MathSciNet  Google Scholar 

  • [So] S. L. Sobolev. 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).

    Article  MATH  Google Scholar 

  • [St1] W. Stolz. Une méthode élémentaire pour l’évaluation de petites boules browniennes. C. R. Acad. Sci. Paris, 316, 1217–1220 (1993).

    MathSciNet  Google Scholar 

  • [St2] W. Stolz. Some small ball probabilities for Gaussian processes under non-uniform norms (1994). To appear in J. Theoretical Prob..

    Google Scholar 

  • [Str] D. Stroock. Homogeneous chaos revisited. Séminaire de Probabilités XXI. Lecture Notes in Math. 1247, 1–7 (1987). Springer-Verlag.

    Article  MathSciNet  MATH  Google Scholar 

  • [Su1] V. N. Sudakov. Gaussian measures, Cauchy measures and ε-entropy. Soviet Math. Dokl. 10, 310–313 (1969).

    MathSciNet  MATH  Google Scholar 

  • [Su2] V. N. Sudakov. Gaussian random processes and measures of solid angles in Hilbert spaces. Soviet Math. Dokl. 12, 412–415 (1971).

    MATH  Google Scholar 

  • [Su3] V. N. Sudakov. A remark on the criterion of continuity of Gaussian sample functions. Proceedings of the Second Japan-USSR Symposium on Probability Theory. Lecture Notes in Math. 330, 444–454 (1973). Springer-Verlag.

    Article  MathSciNet  Google Scholar 

  • [Su4] V. N. Sudakov, Geometric problems of the theory of infinite-dimensional probability distributions. Trudy Mat. Inst. Steklov 141 (1976).

    Google Scholar 

  • [S-T] V. N. Sudakov, B. S. Tsirel’son. 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).

    Article  MATH  Google Scholar 

  • [Sy] G. N. Sytaya. On some asymptotic representation of the Gaussian measure in a Hilbert space. Theory of Stochastic Processes (Kiev) 2, 94–104 (1974).

    MATH  Google Scholar 

  • [Sz] S. Szarek. On the best constant in the Khintchine inequality. Studia Math. 58, 197–208 (1976).

    MathSciNet  MATH  Google Scholar 

  • [Tak] M. Takeda. On a martingale method for symmetric diffusion processes and its applications. Osaka J. Math. 26, 605–623 (1989).

    MathSciNet  MATH  Google Scholar 

  • [Ta1] M. Talagrand. Sur l’intégrabilité des vecteurs gaussiens. Z. Wahrscheinlichkeitstheor. verw. Gebiete 68, 1–8 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  • [Ta2] M. Talagrand. Regularity of Gaussian processes. Acta Math. 159, 99–149 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  • [Ta3] M. Talagrand. An isoperimetric theorem on the cube and the Khintchin-Kahane inequalities. Proc. Amer. Math. Soc. 104, 905–909 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  • [Ta4] M. Talagrand. Small tails for the supremum of a Gaussian process. Ann. Inst. H. Poincaré 24, 307–315 (1988).

    MathSciNet  MATH  Google Scholar 

  • [Ta5] M. Talagrand. Isoperimetry and integrability of the sum of independent Banach space valued random variables. Ann. Probability 17, 1546–1570 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  • [Ta6] M. Talagrand. A new isoperimetric inequality for product measure and the tails of sums of independent random variables. Geometric and Funct. Anal. 1, 211–223 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  • [Ta7] M. Talagrand. Simple proof of the majorizing measure theorem. Geometric and Funct. Anal. 2, 118–125 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  • [Ta8] M. Talagrand. On the rate of clustering in Strassen’s law of the iterated logarithm. Probability in Banach spaces 8. Progress in Probability 30, 339–351 (1992). Birkhäuser.

    MathSciNet  MATH  Google Scholar 

  • [Ta9] M. Talagrand. New Gaussian estimates for enlarged balls. Geometric and Funct. Anal. 3, 502–526 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  • [Ta10] M. Talagrand. Regularity of infinitely divisible processes. Ann. Probability 21, 362–432 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  • [Ta11] M. Talagrand. Isoperimetry, logarithmic Sobolev inequalities on the discrete cube, and Margulis’ graph connectivity theorem. Geometric and Funct. Anal. 3, 295–314 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  • [Ta12] M. Talagrand. The supremum of some canonical processes. Amer. Math. J. 116, 283–325 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  • [Ta13] M. Talagrand. Sharper bounds for Gaussian and empirical processes. Ann. Probability 22, 28–76 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  • [Ta14] M. Talagrand. Constructions of majorizing measures. Bernoulli processes and cotype. Geometric and Funct. Anal. 4, 660–717 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  • [Ta15] M. Talagrand. The small ball problem for the Brownian sheet. Ann. Probability 22, 1331–1354 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  • [Ta16] M. Talagrand. Concentration of measure and isoperimetric inequalities in product spaces (1994). To appear in Publ. de l’IHES.

    Google Scholar 

  • [Ta17] M. Talagrand. Isoperimetry in product spaces: higher level, large sets. Preprint (1994).

    Google Scholar 

  • [Ta18] M. Talagrand. Majorizing measures: the generic chaining. Preprint (1994).

    Google Scholar 

  • [TJ] N. Tomczak-Jaegermann. Dualité des nombres d’entropie pour des opérateurs à valeurs dans un espace de Hilbert. C. R. Acad. Sci. Paris 305, 299–301 (1987).

    MathSciNet  MATH  Google Scholar 

  • [Var] S. R. S. Varadhan. Large deviations and applications. S. I. A. M. Philadelphia (1984).

    Book  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • [Va3] N. Varopoulos. Isoperimetric inequalities and Markov chains. J. Funct. Anal. 63, 215–239 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  • [Va4] N. Varopoulos. Small time Gaussian estimates of heat diffusion kernels. Part I: The semigroup technique. Bull. Sc. math. 113, 253–277 (1989).

    MathSciNet  MATH  Google Scholar 

  • [Va5] N. Varopoulos. Analysis and geometry on groups. Proceedings of the International Congress of Mathematicians, Kyoto (1990), vol. II, 951–957 (1991). Springer-Verlag.

    MathSciNet  MATH  Google Scholar 

  • [V-SC-C] N. Varopoulos, L. Saloff-Coste, T. Coulhon. Analysis and geometry on groups. Cambridge Univ. Press (1992).

    Google Scholar 

  • [Wa] S. Watanabe. Lectures on stochastic differential equations and Malliavin calculus. Tata Institute of Fundamental Research Lecture Notes. Springer-Verlag (1984).

    Google Scholar 

  • [W-W] D. L. Wang, P. Wang. Extremal configurations on a discrete torus and a generalization of the generalized Macaulay theorem. Siam J. Appl. Math. 33, 55–59 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  • [Wi] N. Wiener. The homogeneous chaos. Amer. Math. J. 60, 897–936 (1930).

    Article  MathSciNet  MATH  Google Scholar 

  • [Ya] S.-T. Yau. Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold. Ann. scient. Éc. Norm. Sup. 8, 487–507 (1975).

    MathSciNet  MATH  Google Scholar 

  • [Zo] V. M. Zolotarev. Asymptotic behaviour of the Gaussian measure in ℓ2. J. Sov. Math. 24, 2330–2334 (1986).

    Article  Google Scholar 

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Ledoux, M. (1996). Isoperimetry and Gaussian analysis. In: Bernard, P. (eds) Lectures on Probability Theory and Statistics. Lecture Notes in Mathematics, vol 1648. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0095676

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