Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
[A-C] R. A. Adams, F. H. Clarke. Gross’s logarithmic Sobolev inequality: a simple proof. Amer. J. Math. 101, 1265–1269 (1979).
[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.
[A-M-S] S. Aida, T. Masuda, I. Shigekawa. Logarithmic Sobolev inequalities and exponential integrability. J. Funct. Anal. 126, 83–101 (1994).
[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).
[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).
[A-G] M. Arcones, E. Giné. On decoupling, series expansions and tail behavior of chaos processes. J. Theoretical Prob. 6, 101–122 (1993).
[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.
[Azu] K. Azuma. Weighted sums of certain dependent random variables. Tohoku Math. J. 19, 357–367 (1967).
[B-C] A. Badrikian, S. Chevet. Mesures cylindriques, espaces de Wiener et fonctions aléatoires gaussiennes. Lecture Notes in Math 379, (1974). Springer-Verlag.
[Ba] A. Baernstein II. Integral means, univalent functions and circular symmetrization. Acta Math. 133, 139–169 (1974).
[B-T] A. Baernstein II, B. A. Taylor. Spherical rearrangements, subharmonic functions and *-functions in n-space. Duke Math. J. 43, 245–268 (1976).
[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.
[B-É] D. Bakry, M. Émery. Diffusions hypercontractives. Séminaire de Probabilités XIX. Lecture Notes in Math. 1123, 175–206 (1985). Springer-Verlag.
[B-R] P. Baldi, B. Roynette. Some exact equivalents for Brownian motion in Hölder norm. Prob. Th. Rel. Fields 93, 457–484 (1992).
[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).
[Bas] R. Bass. Probability estimates for multiparameter Brownian processes. Ann. Probability 16, 251–264 (1988).
[Be1] W. Beckner. Inequalities in Fourier analysis. Ann. Math. 102, 159–182 (1975).
[Be2] W. Beckner. Unpublished (1982).
[Be3] W. Beckner. Sobolev inequalities, the Poisson semigroup and analysis on the sphere S n. Proc. Nat. Acad. Sci. 89, 4816–4819 (1992).
[Bel] D. R. Bell. The Malliavin calculus. Pitman Monographs 34. Longman (1987).
[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.
[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.
[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).
[Bob1] S. Bobkov. A functional form of the isoperimetric inequality for the Gaussian measure (1993). To appear in J. Funct. Anal.
[Bob2] S. Bobkov. An isoperimetric inequality on the discrete cube and an elementary proof of the isoperimetric inequality in Gauss space. Preprint (1994).
[Bog] V. I. Bogachev. Gaussian measures on linear spaces (1994). To appear.
[Bon] A. Bonami. Etude des coefficients de Fourier des fonctions de L p (G). Ann. Inst. Fourier 20, 335–402 (1970).
[Bo1] C. Borell. Convex measures on locally convex spaces. Ark. Mat. 12, 239–252 (1974).
[Bo2] C. Borell. The Brunn-Minskowski inequality in Gauss space. Invent. Math. 30, 207–216 (1975).
[Bo3] C. Borell. Gaussian Radon measures on locally convex spaces. Math. Scand. 38, 265–284 (1976).
[Bo4] C. Borell. A note on Gauss measures which agree on small balls. Ann. Inst. H. Poincaré 13, 231–238 (1977).
[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.
[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.
[Bo7] C. Borell. A Gaussian correlation inequality for certain bodies in ℝn. Math. Ann. 256, 569–573 (1981).
[Bo8] C. Borell. On polynomials chaos and integrability. Prob. Math. Statist. 3, 191–203 (1984).
[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).
[Bo10] C. Borell. Geometric bounds on the Ornstein-Uhlenbeck process. Z. Wahrscheinlichkeitstheor. verw. Gebiete 70, 1–13 (1985).
[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.
[Bo12] C. Borell. Analytic and empirical evidence of isoperimetric processes. Probability in Banach spaces 6. Progress in Probability 20, 13–40 (1990). Birkhäuser.
[B-M] A. Borovkov, A. Mogulskii. On probabilities of small deviations for stochastic processes. Siberian Adv. Math. 1, 39–63 (1991).
[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).
[B-Z] Y. D. Burago, V. A. Zalgaller. Geometric inequalities. Springer-Verlag (1988). First Edition (russian): Nauka (1980).
[C-M] R. H. Cameron, W. T. Martin. Transformations of Wiener integrals under translations. Ann. Math. 45, 386–396 (1944).
[Ca] M. Capitaine. Onsager-Machlup functional for some smooth norms on Wiener space (1994). To appear in Prob. Th. Rel. Fields.
[Ca-L] E. Carlen, M. Loss. Extremals of functionals with competing symmetries. J. Funct. Anal. 88, 437–456 (1990).
[C-F] I. Chavel, E. Feldman. Modified isoperimetric constants, and large time heat diffusion in Riemannian manifold. Duke Math. J. 64, 473–499 (1991).
[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).
[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).
[Che] S. Chevet. Gaussian measures and large deviations. Probability in Banach spaces IV. Lecture Notes in Math. 990, 30–46 (1983). Springer-Verlag.
[Ci1] Z. Ciesielski. On the isomorphisms of the spaces H α and m. Bull. Acad. Pol. Sc. 8, 217–222 (1960).
[Ci2] Z. Ciesielski. Orlicz spaces, spline systems and brownian motion. Constr. Approx. 9, 191–208 (1993).
[Co] F. Comets. A spherical bound for the Sherrington-Kirkpatrick model. Preprint (1994).
[C-N] F. Comets, J. Neveu. The Sherrington-Kirkpatrick model of spin glasses and stochastic calculus: the high temperature case. Preprint (1993).
[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).
[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.
[Da] E. B. Davies. Heat kernels and spectral theory. Cambridge Univ. Press (1989).
[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).
[D-L] P. Deheuvels, M. A. Lifshits. Strassen-type functional laws for strong topologies. Prob. Th. Rel. Fields 97, 151–167 (1993).
[De] J. Delporte. Fonctions aléatoires presque sûrement continues sur un intervalle fermé. Ann. Inst. H. Poincaré 1, 111–215 (1964).
[D-S] J.-D. Deuschel, D. Stroock. Large deviations. Academic Press (1989).
[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).
[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).
[Du1] R. M. Dudley. The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Funct. Anal. 1, 290–330 (1967).
[Du2] R. M. Dudley. Sample functions of the Gaussian process. Ann. Probability 1, 66–103 (1973).
[D-HJ-S] R. M. Dudley, J. Hoffmann-Jorgensen, L. A. Shepp. On the lower tail of Gaussianseminorms. Ann. Probability 7, 319–342 (1979).
[Dv] A. Dvoretzky. Some results on convex bodies and Banach spaces. Proc. Symp. on Linear Spaces, Jerusalem, 123–160 (1961).
[Eh1] A. Ehrhard. Une démonstration de l’inégalité de Borell. Ann. Scientifiques de l’Université de Clermont-Ferrand 69, 165–184 (1981).
[Eh2] A. Ehrhard. Symétrisation dans l’espace de Gauss. Math. Scand. 53, 281–301 (1983).
[Eh3] A. Ehrhard. Inégalités isopérimétriques et intégrales de Dirichlet gaussiennes. Ann. scient. Éc. Norm. Sup. 17, 317–332 (1984).
[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.
[Eh5] A. Ehrhard. Eléments extrémaux pour les inégalités de Brunn-Minkowski gaussiennes. Ann. Inst. H. Poincaré 22, 149–168 (1986).
[E-S] O. Enchev, D. Stroock. Rademacher’s theorem for Wiener functionals. Ann. Probability 21, 25–33 (1993).
[Fa] S. Fang. On the Ornstein-Uhlenbeck process. Stochastics and Stochastic Reports 46, 141–159 (1994).
[Fed] H. Federer. Geometric measure theory. Springer-Verlag (1969).
[F-F] H. Federer, W. H. Fleming. Normal and integral current. Ann. Math. 72, 458–520 (1960).
[Fe1] X. Fernique. Continuité des processus gaussiens. C. R. Acad. Sci. Paris 258, 6058–6060 (1964).
[Fe2] X. Fernique Intégrabilité des vecteurs gaussiens. C. R. Acad. Sci. Paris 270, 1698–1699 (1970).
[Fe3] X. Fernique. Régularité des processus gaussiens. Invent. Math. 12, 304–320 (1971).
[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.
[Fe5] X. Fernique. Gaussian random vectors and their reproducing kernel Hilbert spaces. Technical report, University of Ottawa (1985).
[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).
[F-W1] M. Freidlin, A. Wentzell. On small random perturbations of dynamical systems. Russian Math. Surveys 25, 1–55 (1970).
[F-W2] M. Freidlin, A. Wentzell. Random perturbations of dynamical systems. Springer-Verlag (1984).
[Ga] E. Gagliardo. Proprieta di alcune classi di funzioni in piu variabili. Ricerche Mat. 7, 102–137 (1958).
[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).
[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).
[G-H-L] S. Gallot, D. Hulin, J. Lafontaine. Riemannian Geometry. Second Edition. Springer-Verlag (1990).
[Go1] V. Goodman. Characteristics of normal samples. Ann. Probability 16, 1281–1290 (1988).
[Go2] V. Goodman. Some probability and entropy estimates for Gaussian measures. Probability in Banach spaces 6. Progress in Probability 20, 150–156 (1990). Birkhäuser.
[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.
[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).
[Gri] K. Grill. Exact convergence rate in Strassen’s law of the iterated logarithm. J. Theoretical Prob. 5, 197–204 (1991).
[Gro] M. Gromov. Paul Lévy’s isoperimetric inequality. Preprint I.H.E.S. (1980).
[G-M] M. Gromov, V. D. Milman. A topological application of the isoperimetric inequality. Amer. J. Math. 105, 843–854 (1983).
[Gr1] L. Gross. Abstract Wiener spaces. Proc. 5th Berkeley Symp. Math. Stat. Prob. 2, 31–42 (1965).
[Gr2] L. Gross. Potential theory on Hilbert space. J. Funct. Anal. 1, 123–181 (1967).
[Gr3] L. Gross. Logarithmic Sobolev inequalities. Amer. J. Math. 97, 1061–1083 (1975).
[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.
[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.
[Ha] H. Hadwiger. Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Springer-Verlag (1957).
[Har] L. H. Harper. Optimal numbering and isoperimetric problems on graphs. J. Comb. Th. 1, 385–393 (1966).
[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).
[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.
[I-W] N. Ikeda, S. Watanabe. Stochastic differential equations and diffusion processes. North-Holland (1989).
[It] K. Itô. Multiple Wiener integrals. J. Math. Soc. Japan 3, 157–164 (1951).
[Ka1] J.-P. Kahane. Sur les sommes vectorielles ∑±u n . C. R. Acad. Sci. Paris 259, 2577–2580 (1964).
[Ka2] J.-P. Kahane. Some random series of functions. Health Math. Monographs (1968). Second Edition: Cambridge Univ. Press (1985).
[Ke] H. Kesten. On the speed of convergence in first-passage percolation. Ann. Appl. Probability 3, 296–338 (1993).
[Kh] C. Khatri. On certain inequalities for normal distributions and their applications to simultaneous confidence bounds. Ann. Math. Statist. 38, 1853–1867 (1967).
[K-L1] J. Kuelbs, W. Li. Small ball probabilities for Brownian motion and the Brownian sheet. J. Theoretical Prob. 6, 547–577 (1993).
[K-L2] J. Kuelbs, W. Li. Metric entropy and the small ball problem for Gaussian measures J. Funct. Anal. 116, 133–157 (1993).
[K-L-L] J. Kuelbs, W. Li, W. Linde. The Gaussian measure of shifted balls. Prob. Th. Rel. Fields 98, 143–162 (1994).
[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..
[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).
[Ku] H.-H. Kuo. Gaussian measures in Banach spaces. Lecture Notes in Math. 436 (1975). Springer-Verlag.
[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).
[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.
[K-S] S. Kwapień, J. Sawa. On some conjecture concerning Gaussian measures of dilatations of convex symmetric sets. Studia Math. 105, 173–187 (1993).
[L-S] H. J. Landau, L. A. Shepp. On the supremum of a Gaussian process. Sankhyà A32, 369–378 (1970).
[La] R. Latala. A note on the Ehrhard inequality. Preprint (1994).
[L-O] R. Latala, K. Oleszkiewicz. On the best constant in the Khintchine-Kahane inequality. Studia Math. 109, 101–104 (1994).
[Led1] M. Ledoux. Isopérimétrie et inégalités de Sobolev logarithmiques gaussiennes. C. R. Acad. Sci. Paris 306, 79–82 (1988).
[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.
[Led3] M. Ledoux. On an integral criterion for hypercontractivity of diffusion semigroups and extremal functions. J. Funct. Anal. 105, 444–465 (1992).
[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).
[Led5] M. Ledoux. Semigroup proofs of the isoperimetric inequality in Euclidean and Gauss space. Bull. Sci. math. 118, 485–510 (1994).
[L-T1] M. Ledoux, M. Talagrand. Characterization of the law of the iterated logarithm in Banach spaces. Ann. Probability 16, 1242–1264 (1988).
[L-T2] M. Ledoux, M. Talagrand. Probability in Banach spaces (Isoperimetry and processes). Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag (1991).
[Lé] P. Lévy. Problèmes concrets d’analyse fonctionnelle. Gauthier-Villars (1951).
[Li] W. Li. Comparison results for the lower tail of Gaussian semi-norms. J. Theoretical Prob. 5, 1–31 (1992).
[Li-S] W. Li, Q.-M. Shao. Small ball estimates for Gaussian processes under Sobolev type norms. Preprint (1994).
[L-Y] P. Li, S.-T. Yau. On the parabolic kernel of the Schrödinger operator. Acta Math. 156, 153–201 (1986).
[Lif1] M. A. Lifshits. On the distribution of the maximum of a Gaussian process. Probability Theory and its Appl. 31, 125–132 (1987).
[Lif2] M. A. Lifshits. Tail probabilities of Gaussian suprema and Laplace transform. Ann. Inst. H. Poincaré 30, 163–180 (1994).
[Lif3] M. A. Lifshits. Gaussian random functions (1994). Kluwer, to appear.
[Lif-T] M. A. Lifshits, B. S. Tsirel’son. Small deviations of Gaussian fields. Probability Theory and its Appl. 31, 557–558 (1987).
[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).
[MD] C. J. H. McDiarmid. On the method of bounded differeces. Twelfth British Combinatorial Conference. Surveys in Combinatorics, 148–188 (1989). Cambrige Univ. Press.
[MK] H. P. McKean. Geometry of differential space. Ann. Probability 1, 197–206 (1973).
[M-P] M. B. Marcus, G. Pisier. Random Fourier series with applications to harmonic analysis. Ann. Math. Studies, vol. 101 (1981). Princeton Univ. Press.
[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).
[Ma1] B. Maurey. Constructions de suites symétriques. C. R. Acad Sci. Paris 288, 679–681 (1979).
[Ma2] B. Maurey. Sous-espaces ℓp des espaces de Banach. Séminaire Bourbaki, exp. 608. Astérisque 105–106, 199–216 (1983).
[Ma3] B. Maurey. Some deviations inequalities. Geometric and Fucnt. Anal. 1, 188–197 (1991).
[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.
[Maz1] V. G. Maz’ja. Classes of domains and imbedding theorems for function spaces. Soviet Math. Dokl. 1, 882–885 (1960).
[Maz2] V. G. Maz’ja, Sobolev spaces. Springer-Verlag (1985).
[Me] M. Mellouk. Support des diffusions dans les espaces de Besov-Orlicz. C. R. Acad. Sci. Paris 319, 261–266 (1994).
[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.
[Mi1] V. D. Milman. New proof of the theorem of Dvoretzky on sections of convex bodies. Funct. Anal. Appl. 5, 28–37 (1971).
[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).
[Mi3] V. D. Milman. Dvoretzky’s theorem—Thirty years later (Survey). Geometric and Funct. Anal. 2, 455–479 (1992).
[Mi-S] V. D. Milman, G. Schechtman. Asymptotic theory of finite dimensional normed spaces. Lecture Notes in Math. 1200 (1986). Springer-Verlag.
[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.
[Mo] J. Moser. On Harnack’s theorem for elliptic differential equations. Comm. Pure Appl. Math. 14, 557–591 (1961).
[Na] J. Nash. Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80, 931–954 (1958).
[Nel] E. Nelson. The free Markov field. J. Funct. Anal. 12, 211–227 (1973).
[Ne1] J. Neveu. Processus aléatoires gaussiens. Presses de l’Université de Montréal (1968).
[Ne2] J. Neveu. Martingales à temps discret. Masson (1972).
[Ne3] J. Neveu. Sur l’espérance conditionnelle par rapport à un mouvement brownien. Ann. Inst. H. Poincaré 2, 105–109 (1976).
[Ni] L. Nirenberg. On elliptic partial differential equations. Ann. Sc. Norm. Sup. Pisa 13, 116–162 (1959).
[Nu] D. Nualart. The Malliavin calculs and related topics (1994). To appear.
[Os] R. Osserman. The isoperimetric inequality. Bull. Amer. Math. Soc. 84, 1182–1238 (1978).
[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.
[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).
[Pi3] G. Pisier. The volume of convex bodies and Banach space geometry. Cambridge Univ. Press (1989).
[Pit] L. Pitt. A Gaussian correlation inequality for symmetric convex sets. Ann. Probability 5, 470–474 (1977).
[Pr1] C. Preston. Banach spaces arising from some integral inequalities. Indiana Math. J. 20, 997–1015 (1971).
[Pr2] C. Preston. Continuity properties of some Gaussian processes. Ann. Math. Statist. 43, 285–292 (1972).
[Sc] M. Schilder. Asymptotic formulas for Wiener integrals. Trans. Amer. Math. Soc. 125, 63–85 (1966).
[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).
[Sco] A. Scott. A note on conservative confidence regions for the mean value of multivariate normal. Ann. Math. Statist. 38, 278–280 (1967).
[S-Z1] L. A. Shepp, O. Zeitouni. A note on conditional exponential moments and Onsager-Machlup functionals. Ann. Probability 20, 652–654 (1992).
[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.
[Sh] Q.-M. Shao. A note on small ball probability of a Gaussian process with stationary increments. J. Theoretical Prob. 6, 595–602 (1993).
[S-W] Q.-M. Shao, D. Wang. Small ball probabilities of Gaussian fields. Preprint (1994).
[Si] Z. Sidak. Rectangular confidence regions for the means of multivariate normal distributions. J. Amer. Statist. Assoc. 62, 626–633 (1967).
[Sk] A. V. Skorohod. A note on Gaussian measures in a Banach space. Theor. Probability Appl. 15, 519–520 (1970).
[Sl] D. Slepian. The one-sided barrier problem for Gaussian noise. Bell. System Tech. J. 41, 463–501 (1962).
[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).
[St1] W. Stolz. Une méthode élémentaire pour l’évaluation de petites boules browniennes. C. R. Acad. Sci. Paris, 316, 1217–1220 (1993).
[St2] W. Stolz. Some small ball probabilities for Gaussian processes under non-uniform norms (1994). To appear in J. Theoretical Prob..
[Str] D. Stroock. Homogeneous chaos revisited. Séminaire de Probabilités XXI. Lecture Notes in Math. 1247, 1–7 (1987). Springer-Verlag.
[Su1] V. N. Sudakov. Gaussian measures, Cauchy measures and ε-entropy. Soviet Math. Dokl. 10, 310–313 (1969).
[Su2] V. N. Sudakov. Gaussian random processes and measures of solid angles in Hilbert spaces. Soviet Math. Dokl. 12, 412–415 (1971).
[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.
[Su4] V. N. Sudakov, Geometric problems of the theory of infinite-dimensional probability distributions. Trudy Mat. Inst. Steklov 141 (1976).
[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).
[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).
[Sz] S. Szarek. On the best constant in the Khintchine inequality. Studia Math. 58, 197–208 (1976).
[Tak] M. Takeda. On a martingale method for symmetric diffusion processes and its applications. Osaka J. Math. 26, 605–623 (1989).
[Ta1] M. Talagrand. Sur l’intégrabilité des vecteurs gaussiens. Z. Wahrscheinlichkeitstheor. verw. Gebiete 68, 1–8 (1984).
[Ta2] M. Talagrand. Regularity of Gaussian processes. Acta Math. 159, 99–149 (1987).
[Ta3] M. Talagrand. An isoperimetric theorem on the cube and the Khintchin-Kahane inequalities. Proc. Amer. Math. Soc. 104, 905–909 (1988).
[Ta4] M. Talagrand. Small tails for the supremum of a Gaussian process. Ann. Inst. H. Poincaré 24, 307–315 (1988).
[Ta5] M. Talagrand. Isoperimetry and integrability of the sum of independent Banach space valued random variables. Ann. Probability 17, 1546–1570 (1989).
[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).
[Ta7] M. Talagrand. Simple proof of the majorizing measure theorem. Geometric and Funct. Anal. 2, 118–125 (1992).
[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.
[Ta9] M. Talagrand. New Gaussian estimates for enlarged balls. Geometric and Funct. Anal. 3, 502–526 (1993).
[Ta10] M. Talagrand. Regularity of infinitely divisible processes. Ann. Probability 21, 362–432 (1993).
[Ta11] M. Talagrand. Isoperimetry, logarithmic Sobolev inequalities on the discrete cube, and Margulis’ graph connectivity theorem. Geometric and Funct. Anal. 3, 295–314 (1993).
[Ta12] M. Talagrand. The supremum of some canonical processes. Amer. Math. J. 116, 283–325 (1994).
[Ta13] M. Talagrand. Sharper bounds for Gaussian and empirical processes. Ann. Probability 22, 28–76 (1994).
[Ta14] M. Talagrand. Constructions of majorizing measures. Bernoulli processes and cotype. Geometric and Funct. Anal. 4, 660–717 (1994).
[Ta15] M. Talagrand. The small ball problem for the Brownian sheet. Ann. Probability 22, 1331–1354 (1994).
[Ta16] M. Talagrand. Concentration of measure and isoperimetric inequalities in product spaces (1994). To appear in Publ. de l’IHES.
[Ta17] M. Talagrand. Isoperimetry in product spaces: higher level, large sets. Preprint (1994).
[Ta18] M. Talagrand. Majorizing measures: the generic chaining. Preprint (1994).
[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).
[Var] S. R. S. Varadhan. Large deviations and applications. S. I. A. M. Philadelphia (1984).
[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).
[Va2] N. Varopoulos. Hardy-Littlewood theory for semigroups. J. Funct. Anal. 63, 240–260 (1985).
[Va3] N. Varopoulos. Isoperimetric inequalities and Markov chains. J. Funct. Anal. 63, 215–239 (1985).
[Va4] N. Varopoulos. Small time Gaussian estimates of heat diffusion kernels. Part I: The semigroup technique. Bull. Sc. math. 113, 253–277 (1989).
[Va5] N. Varopoulos. Analysis and geometry on groups. Proceedings of the International Congress of Mathematicians, Kyoto (1990), vol. II, 951–957 (1991). Springer-Verlag.
[V-SC-C] N. Varopoulos, L. Saloff-Coste, T. Coulhon. Analysis and geometry on groups. Cambridge Univ. Press (1992).
[Wa] S. Watanabe. Lectures on stochastic differential equations and Malliavin calculus. Tata Institute of Fundamental Research Lecture Notes. Springer-Verlag (1984).
[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).
[Wi] N. Wiener. The homogeneous chaos. Amer. Math. J. 60, 897–936 (1930).
[Ya] S.-T. Yau. Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold. Ann. scient. Éc. Norm. Sup. 8, 487–507 (1975).
[Zo] V. M. Zolotarev. Asymptotic behaviour of the Gaussian measure in ℓ2. J. Sov. Math. 24, 2330–2334 (1986).
Editor information
Rights and permissions
Copyright information
© 1996 Springer-Verlag
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/BFb0095676
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-62055-6
Online ISBN: 978-3-540-49635-9
eBook Packages: Springer Book Archive