Skip to main content
SpringerLink
Log in

Probability and geometry

  • Conference paper
  • First Online:
Probability and Analysis

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1206))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. ALVAREZ-GAUME L.: Supersymmetry and the Atiyah-Singer Index Theorem. Commun. Math. Physics 90(1983), 161–173.

    Article  MATH  MathSciNet  Google Scholar 

  2. ATIYAH M.F.: K-Theory. Benjamin 1967.

    Google Scholar 

  3. ATIYAH M.F.: Classical groups and classical operators on manifolds. In "Differential operators manifolds". CIME, Cremonese, Rome 1975.

    Google Scholar 

  4. ATIYAH M.F.: Circular symmetry and stationary phase approximation. In "Proceedings of the Conference in Honor of L. Schwartz". Astérisque 131 (1985), 43–59

    MathSciNet  Google Scholar 

  5. ATIYAH M.F., BOTT R.: A Lefschetz fixed point formula for elliptic complexes. I, Ann. of Math. 86 (1967), 394–407; II, 88(1968), 451–491.

    Article  MathSciNet  Google Scholar 

  6. ATIYAH M.F., BOTT R.: The moment map and equivariant cohomology. Topology 23,1–28(1984).

    Article  MATH  MathSciNet  Google Scholar 

  7. ATIYAH M.F., BOTT R., PATODI V.K.: On the heat equation and the Index Theorem. Invent. Math. 19,279–330(1973).

    Article  MATH  MathSciNet  Google Scholar 

  8. ATIYAH M.F., SINGER I.M.: The Index of elliptic operators. I, Ann. of Math. 87(1968), 485–530; III, Ann. of Math. 87(1968), 546–604; IV, Ann. of Math. 93(1971), 119–138.

    Google Scholar 

  9. AZENCOTT R.: Grandes déviations et applications. Cours de probabilité de Saint-Flour. Lecture Notes in Math. no 774. Berlin Springer 1978.

    Google Scholar 

  10. AZENCOTT R.: Densités des diffusions en temps petit: développements en temps petit. In Séminaire de Probabilités no XVIII, p. 402–498. Lecture Notes Math. no 1059, Berlin, Springer 1984.

    Google Scholar 

  11. AZENCOTT R.: Une approche probabiliste du Théorème d'Atiyah-Singer, d'après J.M. Bismut. Séminaire Bourbaki 1984–1985. Astérsque 133–134 (1986), 7–18

    MathSciNet  Google Scholar 

  12. AZENCOTT R., BALDI P., BELLAICHE A., BELLAICHE C., BOUGEROL P., CHALEYAT-MAUREL M., ELIE L., GRANARA J.: Géodésiques et diffusions en temps petit. Astérisque 84–85. Paris 1981.

    Google Scholar 

  13. BERLINE N., VERGNE M.: Zéros d'un champ de vecteurs et classes caractéristiques équivariantes. Duke Math. J. 50, 539–549(1983).

    Article  MATH  MathSciNet  Google Scholar 

  14. BERLINE N., VERGNE M.: The equivariant index and Kirillov's character formula. Am. J. Math. 107 (1985), 1159–1190

    Article  MATH  MathSciNet  Google Scholar 

  15. BERLINE N., VERGNE M.: A computation of the equivariant index of the Dirac operator. Bull. Soc. Math. Fr. 113 (1985), 305–345

    MATH  MathSciNet  Google Scholar 

  16. BISMUT J.M.: Mécanique aléatorie. Lecture Notes in Math. no 866. Berlin: Springer 1981.

    Google Scholar 

  17. BISMUT J.M.: Large deviations and the Malliavin calculus. Progress in Math. no 45. Birkhäuser 1984.

    Google Scholar 

  18. BISMUT J.M.: Transformations différentiables du mouvement Brownien. Proceedings of the Conference in honor of L. Schwartz. Astérisque 131 Paris 1985. 61–87

    Google Scholar 

  19. BISMUT J.M.: The Atiyah-Singer Theorems: a probabilistic approach. I, The Index Theorem. J. Funct. Anal. 57(1984). 56–99. II, The Lefschetz fixed-point formulas.J. Funct. Anal. 57(1984), 329–348.

    Article  MATH  MathSciNet  Google Scholar 

  20. BISMUT J.M.: Index Theorem and Equivariant Cohomology on the loop space. Comm. in Math. Physics. 98,213–237(1985).

    Article  MATH  MathSciNet  Google Scholar 

  21. BISMUT J.M.: The infinitesimal Lefschetz formulas: A heat equation proof. J. Funct. Anal. 62 (1985), 435–457

    Article  MATH  MathSciNet  Google Scholar 

  22. BISMUT J.M.: Formules de Lefschetz délocalisées. Séminaire Bony-Sjöstrand-Meyer 1984–1985. Exp. 7. 13pp. Ecole Polytechn. Palaiseau (1985)

    Google Scholar 

  23. BISMUT J.M. The Witten complex and degenerate Morse inequalities. To appear in J. Diff. Geom. (May 1986)

    Google Scholar 

  24. BISMUT J.M.: The Atiyah-Singer Index Theorem for families of Dirac operators: two heat equation proofs. Invent. Math. 83 (1986), 91–151

    Article  MATH  MathSciNet  Google Scholar 

  25. BISMUT J.M. Localization formulas, superconnections and the Index Theorem for families. Comm. Math. Phys. 103 (1986), 127–166

    Article  MATH  MathSciNet  Google Scholar 

  26. BLAGOVESHCHENSKII Y.N., FREIDLIN M.I.: Certain properties of processes depending on parameters. Sov. Math. Dokl. 2(1961), 633–636.

    MATH  Google Scholar 

  27. DE WITT-MORETTE C., MAHESHWARI A., NELSON B.: Path Integration in non relativistic quantum mechanics. Physics Reports 50(1979), 255–372.

    Article  MathSciNet  Google Scholar 

  28. DUISTERMAAT J.J., HECKMAN G.J.: On the variation of the cohomology of the symplectic form of the reduced phase space. Invent. Math. 69(1982), 259–268; Addendum 72(1983),153–158.

    Article  MATH  MathSciNet  Google Scholar 

  29. ELWORTHY K.D., TRUMAN A.: Classical mechanics, the diffusion heat equation and the Schrödinger equation on a Riemannian manifold. J. Math. Phys. 22, 2144–2166(1981).

    Article  MATH  MathSciNet  Google Scholar 

  30. FEFFERMAN C.: The uncertainty principle. Bull AMS 9(1983), 129–206.

    Article  MATH  MathSciNet  Google Scholar 

  31. FRIEDAN D., WINDEY H.: Supersymmetric derivation of the Atiyah-Singer Index and the Chiral anomaly. Nuclear Physics B. 235, (1984)395–416.

    Article  MathSciNet  Google Scholar 

  32. FRIEDMAN A.: Partial differential equation of parabolic type. Englewood-Cliffs. Prentice Hall 1964.

    Google Scholar 

  33. GARSIA A.M., RODEMICH E., RUMSEY H.: A real variable lemma and the continuity of paths of some Gaussian processes. Indiana Univ. Math. J. 20(1970), 565–578.

    Article  MATH  MathSciNet  Google Scholar 

  34. GAVEAU B.: Principe de moindre action, propagation de la chaleur et estimées sous-elliptiques sur certains groupes nilpotents. Acta Math. 139(1977), 96–153.

    Article  MathSciNet  Google Scholar 

  35. GETZLER E.: Pseudodifferential operators on supermanifolds and the Atiyah-Singer Index Theorem. Commun. Math. Phys. 92(1983), 163–178.

    Article  MATH  MathSciNet  Google Scholar 

  36. GETZLER E.: A Short proof of the Atiyah-Singer Index Theorem. Topology 25 (1986), 111–117

    Article  MATH  MathSciNet  Google Scholar 

  37. GILKEY P.: Curvature and the eigenvalues of the Laplacian. Advan. in Math. 10(1973), 344–382.

    Article  MATH  MathSciNet  Google Scholar 

  38. HELFFER B., SJÖSTRAND J.: Sur le complexe de Witten. To appear.

    Google Scholar 

  39. HÖRMANDER L.: Hypoelliptic second order differential equations Acta Math. 119(1967), 147–171.

    Article  MATH  MathSciNet  Google Scholar 

  40. JEULIN T.: Semi-martingales et grossissement d'une filtration. Lecture Notes in Math. no 833. Berlin Springer 1980.

    Google Scholar 

  41. JONA-LASINIO G., MARTINELLI F. SCOPPOLA E.: New approach to the semiclassical limit of Quantum Mechanics. Commun. Math. Phys. 80, 223–254(1981).

    Article  MATH  MathSciNet  Google Scholar 

  42. KIFER Y.: On the asymptotics of the transition density of processes with small diffusion. Theory Prob. and Appl. 21,512–522(1976).

    Google Scholar 

  43. LEANDER R.: To appear.

    Google Scholar 

  44. MALLIAVIN P.: Stochastic calculus of variations and hypoelliptic operators. In "Proceedings of the Conference on Stochastic differential equations of Kyoto", 1976 (K. Itô ed.) pp 155–263. Wiley, New-York 1978.

    Google Scholar 

  45. Mc KEAN H., SINGER I.M.: Curvature and the eigenvalues of the Laplacian. J. of Diff. Geom. 1(1967),43–69.

    MathSciNet  Google Scholar 

  46. MOLCHANOV S.A.: Diffusion processes and Riemannian geometry. Russian Math. Surveys 30(1975),1–53.

    Article  MATH  Google Scholar 

  47. PATODI V.K.: Curvature and the eigenforms of the Laplace operator. J. of Diff. Geom. 5(1971),251–283.

    MATH  MathSciNet  Google Scholar 

  48. PATODI V.K.: An analytic proof of the Riemann-Roch-Hirzehruch Theorem for Kaehler manifolds. J. of Diff. Geom. 5(1971),251–283.

    MATH  MathSciNet  Google Scholar 

  49. QUILLEN D.: Superconnections and the Chern character. Topology 24 (1985), 89–95

    MATH  MathSciNet  Google Scholar 

  50. STROOCK D., VARADHAN S.R.S.: On the support of diffusion processes. 6th Berkeley Symposium on probability and statistics. Vol. III p. 333–359. Berkeley: Univ. of Calif. Press. 1972.

    Google Scholar 

  51. SANCHEZ-CALLE A.: Fundamental solutions and Geometry of the sum of square of vectors fields. Invent. Math. 78(1984), 143–160.

    Article  MATH  MathSciNet  Google Scholar 

  52. VARADHAN S.R.S.: diffusion processes on a small time interval. Commun. Pure and Appl. Math. 20(1967),659–685.

    Article  MATH  MathSciNet  Google Scholar 

  53. VARADHAN S.R.S.: Asymptotic probabilitiesand differential equations Commun. Pure and Appl. Math. XIX(1966),261–286.

    Article  MathSciNet  Google Scholar 

  54. VENTCELL A.D., FREIDLIN M.I.: On small random perturbations of dynamical systems. Russian Math. Surveys 25(1970), p.1–55.

    Article  Google Scholar 

  55. VENTCELL A.D., FREIDLIN M.I.: Random perturbation of dynamical systems. Grundl. Math. Wiss. 260. Berlin: Springer 1984.

    Google Scholar 

  56. WITTEN E.: Supersymmetry and Morse theory. J. of Diff. Geom. 17(1982),661–692.

    MATH  MathSciNet  Google Scholar 

  57. YOR M.: Remarques sur une formule de P. Lévy. In Séminaire de Probabilités noXIV pp. 343–346. Lecture Notes in Math. no784. Berlin: Springer 1980.

    Google Scholar 

  58. HÖRMANDER L.: The analysis of Linear Partial differential operators. Grundl. Math. Wiss. no256. Berlin: Springer 1983.

    Book  Google Scholar 

  59. KUSUOKA S. To appear.

    Google Scholar 

  60. LICHNEROWICZ A.: Spineurs harmoniques. CRAS. Math. 257,7.9(1963).

    Google Scholar 

  61. MALLIAVIN P.: Géométrie différentielle Stochastique. Presses de l'Université de Montréal: Montréal (1978).

    Google Scholar 

  62. TREVES F.: Introduction to pseudodifferential operators and Fourier Integral operators. Vol. 1 New-York: Plenum Press 1980.

    MATH  Google Scholar 

  63. EELLS J., ELWORTHY K.D.: Wiener integration on certain manifolds of maps.In "Some problems in nonlinear analysis". CIME IV, pp. 69–94. Cremonese, Rome 1971.

    Google Scholar 

  64. SMALE S.: Differentiable dynamical systems. Bull Am. Math. Soc. 7 (1967), 747–817.

    Article  MathSciNet  Google Scholar 

  65. VENTCELL A.D.: On the asymptotics of eigenvalues of matrices. Sov. Math.Dokl. 13(1972),65–68.

    Google Scholar 

  66. ITÔ K., Mc KEAN H.: Diffusion processes and their sample paths. Grundl. Math. Wiss 125 Berlin: Springer 1974.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Départment de Mathématique, Université Paris Sud, Bâtiment 425, 91405, Orsay, France

    Jean-Michel Bismut

Authors
  1. Jean-Michel Bismut
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica, Università di Pisa, Via Buonarroti 2, 56100, Pisa, Italy

    Giorgio Letta  & Maurizio Pratelli  & 

Rights and permissions

Reprints and permissions

Copyright information

© 1986 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Bismut, JM. (1986). Probability and geometry. In: Letta, G., Pratelli, M. (eds) Probability and Analysis. Lecture Notes in Mathematics, vol 1206. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0076299

Download citation

  • DOI: https://doi.org/10.1007/BFb0076299

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-16787-7

  • Online ISBN: 978-3-540-40955-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation