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Probabilistic methods in the study of asymptotics

Part of the Lecture Notes in Mathematics book series (LNMECOLE,volume 1427)

Keywords

  • Vector Field
  • Asymptotic Expansion
  • Stochastic Differential Equation
  • Heat Kernel
  • Wiener Space

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. H. Airault, Perturbations singulières et solutions stochastiques de problèmes de D. Neumann-Spencer, J. Math. Pures Appl., 55 (1976), 233–268.

    MathSciNet  MATH  Google Scholar 

  2. H. Airault and P. Malliavin, Intégration géométrique sur l'espace de Wiener, Bull. Sc. Math., 2° series, 112 (1988), 3–52.

    MathSciNet  MATH  Google Scholar 

  3. S. Albeverio and R. Höegh-Krohn, Oscillatory integrals and the method of stationary phase in infinitely many dimensions, with applications to the classical limit of quantum mechanics, I, Inv. Math., 40 (1977), 59–106.

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. M. F. Atiyah, Circular symmetry and stationary-phase approximation, Astérisque, Société Mathématique de France, 131 (1985), 43–59.

    MathSciNet  MATH  Google Scholar 

  5. J. Avron, I. Herbst and B. Simon, Schrödinger operators with magnetic fields, I, General interactions, Duke Math. J., 45 (1978), 847–884.

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. J. Avron, I. Herbst and B. Simon, Schrödinger operators with magnetic fields, II, Separation of the center of mass in homogeneous magnetic fields, Ann. Phys., 114 (1978), 431–451.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. J. Avron, I. Herbst and B. Simon, Schrödinger operators with magnetic fields, III, Atoms in homogeneous magnetic fields, Comm. Math. Phys., 79 (1981), 529–572.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. R. Azencott, “Grandes deviations et applications,” Ecole d'Eté de Probabilités de Saint-Flour VIII-1978, ed. P. L. Hennequin, Lect. Notes in Math.,774 (1980), Springer-Verlag, pp. 2–176.

    Google Scholar 

  9. R. Azencott et al., Géodésiques et diffusions en temps petit, Séminaire de probabilités, Université de Paris VII, Astérisque, Société Mathématique de France, 84–85 (1981).

    Google Scholar 

  10. R. Azencott, “Densités des diffusions en temps petit: dévelopments asymptotics,” Séminaire de Prob., XVIII, 1982–1983, Lect. Notes in Math., 1059, Springer Verlag, pp. 402–498.

    Google Scholar 

  11. Ben Arous, “Noyau de la chaleur hypoelliptique et géométrie sous-riemanniee,” Stochastic Analysis, Proc. Japanese-French Seminar 1987, (ed. by M. Métivier and S. Watanabe), Lect. Notes in Math., Springer-Verlag, 1988, pp. 1–16.

    Google Scholar 

  12. Ben Arous, Methodes de Laplace et de la phase stationaire sur l'espace de Wiener, preprint.

    Google Scholar 

  13. M. Sh. Birman and M. Z. Solomyak, Asymptotic behavior of the spectrum of differential equations, J. Soviet Math., 12 (1979), 247–283.

    CrossRef  MATH  Google Scholar 

  14. J. M. Bismut, “Large deviations and the Malliavin calculus,” Progress in Math., 45, Birkhäuser, 1984.

    Google Scholar 

  15. J. M. Bismut, The Atiyah-Singer theorem, A probabilistic approach 1, The index theorem, 2. The Lefschetz fixed point formula, J. Funct. Anal., 57 (1984), 56–99, 329–348.

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. J. M. Bismut, Index theorem and equivariant cohomology on the loop space, Comm. Math. Phys., 98 (1985), 213–237.

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. J. M. Bismut, The Witten complex and the degenerate Morse inequalities, J. Diff. Geometry, 23 (1986), 207–240.

    MathSciNet  MATH  Google Scholar 

  18. J. M. Bismut, The Atiyah-Singer index theorem for families of Dirac operators; two heat equations proofs, Inven. Math., 83 (1986), 91–151.

    CrossRef  MathSciNet  MATH  Google Scholar 

  19. J. M. Bismut, Localization formulas, superconnections and the index theorem for families, Comm. Math. Phys., 103 (1986), 127–166.

    CrossRef  MathSciNet  MATH  Google Scholar 

  20. R. P. Boas, “Entire functions,” Academic Press, 1954.

    Google Scholar 

  21. V. S. Buslaev, “Continum integrals and the asymptotic behavior of the solution of parabolic equation as t → ∞, Application to diffraction,” Topics in Math. Phys. 2, ed. by M. Sh. Birman, 1968, pp. 67–86.

    Google Scholar 

  22. S. S. Chern, “Selected papers,” Springer-Verlag, 1979.

    Google Scholar 

  23. Y Colin de Verdière, Sectre de Laplacien et longeurs des géodésiques périodiques, I, II, Compositio Math., 27 (1973), 83–106, 159–184.

    MathSciNet  Google Scholar 

  24. Y. Colin de Verdière, L'asymptotique de Weyl pour les bouteilles magnetiques, Comm. Math. Phys., 105 (1986), 327–335.

    CrossRef  MathSciNet  MATH  Google Scholar 

  25. H. L. Cycon, R. G. Froese, W. Kirsch and B. Simon, “Schrödinger operators, with applications to quantum mechanics and global geometry,” Springer-Verlag, 1987.

    Google Scholar 

  26. A. Debiard, B. Gaveau and E. Mazet, Théorémes de comparaison en géométrie riemannienne, Publ. RIMS, Kyoto Univ., 12 (1976), 391–425.

    CrossRef  MathSciNet  MATH  Google Scholar 

  27. J. D. Deuschel and D. W. Stroock, “Large deviations,” Academic Press, 1989.

    Google Scholar 

  28. B. A. Dubrovin, A. T. Fomenko and S. P. Novikov, “Modern geometry-Methods and Applications, Part I, The geometry of surfaces, Transformation groups, and fields, Part II, The geometry and topology of manifolds,” Springer-Verlag, 1984.

    Google Scholar 

  29. A. Dufresonoy, Un example de champ magnetique dans R v, Duke Math. J., 50 (1983), 729–734.

    CrossRef  MathSciNet  Google Scholar 

  30. K. D. Elworthy, “Stochastic differential equations on manifolds,” London Math. Soc. Lect. Note Series, 70, Cambridge Univ. Press, 1982.

    Google Scholar 

  31. K. D. Elworthy, “Geometric aspects of diffusions on manifolds,” Ecole d'Eté de Probabilités de Saint Flour, 16, 1987, to appear in Lect. Notes in Math., Springer-Verlag.

    Google Scholar 

  32. K. D. Elworthy and A. Truman, “The diffusion equation and classical mechanics: an elementary formula, stochastic processes in Quantum Physics,” (ed. by S. Albeverio et al.), Lect. Notes in Phys., 173 (1982), Springer-Verlag, pp. 136–146.

    Google Scholar 

  33. R. P. Feynman and A. R. Hibbs, “Quantum mechanics and path integrals,” McGraw-Hill, 1965.

    Google Scholar 

  34. M. I. Freidlin and A. D. Wentzell, “Random perturbation of dynamical systems,” A series of Comprehensive Studies in Math., 260, Springer-Verlag, 1983.

    Google Scholar 

  35. B. Gaveau, Principle de moindre action, propagation de la chleau et estiméss sous elliptiques sur certain group nilpotents, Acta Math., 136 (1977), 95–153.

    CrossRef  MathSciNet  Google Scholar 

  36. B. Gaveau and J. Vanthier, Intégrales oscillantes stochastiques l'equation de Pauli, J. Funct. Anal., 44 (1981), 388–400.

    CrossRef  MathSciNet  MATH  Google Scholar 

  37. B. Gaveau and J. M. Moulinier, Intégrales oscillantes stochastiques: Estimation asymptotique de fonctionnelles caractéristiques, J. Funct. Anal., 54 (1983), 161–176.

    CrossRef  MathSciNet  MATH  Google Scholar 

  38. E. Getzler, Degree theory for Wiener maps, J. Funct. Anal., 68 (1986), 388–403.

    CrossRef  MathSciNet  MATH  Google Scholar 

  39. E. Getzler, A short proof of the Atiyah-Singer index theorem, Topology, 25 (1986), 111–117.

    CrossRef  MathSciNet  MATH  Google Scholar 

  40. I. M. Gelfand and G. E. Shilov, “Generalized functions,” Vol. 1, Academic Press, 1964.

    Google Scholar 

  41. I. M. Gelfand and N. Ya Vilenkin, “Generalized functions,” Vol. 4, Academic Press, 1964.

    Google Scholar 

  42. P. B. Gilkey, “Invariance theory, the heat equation, and the Atiyah-Singer index theorem,” Math. Lecture Series, 11, Publish or Perish, Inc., 1984.

    Google Scholar 

  43. B. Helffor, “Semi-classical analysis for the Schrödinger operator and applications,” Lect. Notes in Math., 1336 (1988), Springer-Verlag.

    Google Scholar 

  44. T. Hida, “Brownian motion,” Springer-Verlag, 1980.

    Google Scholar 

  45. L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147–171.

    CrossRef  MathSciNet  MATH  Google Scholar 

  46. L. Hörmander, “The analysis of linear partial differential operators I,” Springer-Verlag, 1983.

    Google Scholar 

  47. P. Hsu, Short time asymptotics of the heat kernel on concave boundary, SIAM J. Math. Anal., 20 (1989), 1109–1127.

    CrossRef  MathSciNet  MATH  Google Scholar 

  48. N. Ikeda, “On the asymptotic behavior of the fundamental solution of the heat equation on certain manifolds,” Proc. Taniguchi Intern. Symp. on Stochastic Analysis, Katata-Kyoto, 1982, (ed. by K. Itô), Kinokuniya/North-Holland, 1984, pp. 169–195.

    Google Scholar 

  49. N. Ikeda, “Limit theorems for a class of random currents,” Proc. Taniguchi Intern. Symp. on Probab. methods in Math. Phys., Katata-Kyoto, 1985, (ed. by K. Itô and N. Ikeda), Kinokuniya/Academic Press, 1987, pp. 181–193.

    Google Scholar 

  50. N. Ikeda and S. Kusuoka, “Short time asymptotics for fundamental solution of diffusion equations,” Proc. Japanese-French Seminar, 1987, (ed. by M. Métivier and S. Watanabe), Lect. Notes in Math., 1322 (1989), Springer-Verlag, pp. 37–49.

    Google Scholar 

  51. N. Ikeda and S. Manabe, Integral of differential forms along the path of diffusion processes, Publ. RIMS, Kyoto Univ., 15 (1979), 827–852.

    CrossRef  MathSciNet  MATH  Google Scholar 

  52. N. Ikeda and Y. Ochi, “Central limit theorems and random currents,” Lect. Notes in Contr. and Inform., 78 (1986), Springer-Verlag, pp. 195–205.

    Google Scholar 

  53. N. Ikeda, I. Shigekawa and S. Taniguchi, “The Malliavin calculus and long time asymptotics of certain Wiener integrals,” Proc. Center for Math. Analy. Australian National Univ., 9 (1985), pp. 46–113.

    MathSciNet  MATH  Google Scholar 

  54. N. Ikeda and S. Watanabe, “An introduction to Malliavin's calculus,” Proc. Taniguchi Intern. Symp. on Stochastic Analysis, Katata-Kyoto, 1982, (ed. by K. Itô), Kinokuniya/North-Holland, 1984, pp. 1–52.

    Google Scholar 

  55. N. Ikeda and S. Watanabe, “Malliavin calculus of Wiener functionals and its applications, From local time to global geoometry, control and physics,” Pitman Research Notes in Math. Series, 150, 1986, pp. 132–178.

    MathSciNet  Google Scholar 

  56. N. Ikeda and S. Watanabe, “Stochastic differential equations and diffusion processes,” Second Edition, Kodansha/North-Holland, 1989.

    Google Scholar 

  57. K. Itô, “Lectures on Stochastic Processes,” Tata Inst. Fund. Research, 1968.

    Google Scholar 

  58. K. Itô, “Foundation of stochastic differential equations in infinite dimensional spaces,” CBMS-NSF, Regional Conf. Series in Appl. Math., 1984.

    Google Scholar 

  59. K. Itô and M. Nisio, On the convergence of sums of independent Banach space valued random variables, Osaka J. Math., 5 (1968), 35–48.

    MathSciNet  MATH  Google Scholar 

  60. A. Iwatsuka, The essential spectrum of two-dimensional Schrödinger operators with perturbed constant magnetic fields, J. Math. Kyoto Univ., 23 (1983), 475–480.

    MathSciNet  MATH  Google Scholar 

  61. A. Iwatsuka, Examples of absolutely continuous Schrödinger operators in magnetic fields, Publ. RIMS, Kyoto Univ., 21 (1985), 385–401.

    CrossRef  MathSciNet  MATH  Google Scholar 

  62. A. Iwatsuka, Magnetic Schrödinger operators with compact resolvent, J. Math. Kyoto Univ., 26 (1986), 357–374.

    MathSciNet  MATH  Google Scholar 

  63. M. Kac, “On some connections between probability theory and differential and integral equations,” Proc. 2nd Berkeley Symposium Math. Stat. Prob. (1951), pp. 189–215.

    Google Scholar 

  64. M. Kac, Can one hear the shape of a drum?, Amer. Math., Monthly, 73 (1966), 37–49.

    CrossRef  MathSciNet  MATH  Google Scholar 

  65. M. Kac, “Probability, Number theory, and Statistical Physics,” Selected Papers, MIT Press, 1979.

    Google Scholar 

  66. M. Kac, “Integration in function spaces and some of its applications,” Accademia Nazionale dei Lincei Scuola Normale Superiore, Pisa, 1980.

    MATH  Google Scholar 

  67. Y. Kannai, Off diagonal short time asymptotics for fundamental solutions of diffusion equations, Comm. Partial Diff. Equat., 2 (1970), 781–830.

    CrossRef  MathSciNet  MATH  Google Scholar 

  68. A. Katsuda and T. Sunada, Homology and closed geodesics in a compact Riemannian surface, Amer. J. Math., 109 (1987), 145–156.

    MathSciNet  MATH  Google Scholar 

  69. S. Kobayashi and K. Nomizu, “Foundation of differential geometry, I, II,” Interscience, 1963 and 1969.

    Google Scholar 

  70. T. Kotake, The fixed point theorem of Atiyah-Bott via parabolic operators, Comm. Pure. Appl. Math., 22 (1969), 789–806.

    CrossRef  MathSciNet  MATH  Google Scholar 

  71. S. Kusuoka, “The generalized Malliavin calculus based on Brownian sheet and Bismut's expansions for large deviasions,” Stochastic Processes-Mathematics and Physics, Proc. 1st BiBoS Symp., Lect. Notes in Math., 1158 (1984), SpringerVerlag, pp. 141–157.

    Google Scholar 

  72. S. Kusuoka, “Degree theorem in certain Wiener Riemannian manifolds,” Stochastic Analysis, Proc. Japanese-French Seminar, 1987 (ed. by M. Métivier and S. Watanabe), Lect. Notes in Math., 1322 (1988), pp. 93–108.

    Google Scholar 

  73. S. Kusuoka, “Some remarks on Getzler's degree theorem,” Probability theory and mathematical statistics, Proc. Fifth Japan-USSR Symp. on Probab., Kyoto, 1986 (ed. by S. Watanabe and Y. V. Prohorov), Lect. Notes in Math., 1299 (1988), Springer-Verlag.

    Google Scholar 

  74. S. Kusuoka, “On the foundations of Wiener-Riemannian manifolds,” Stochastic analysis, path integration and dynamics, (ed. by K. D. Elworthy and J. C. Zambrini), Pitman Research Notes in Math. Series, 200 (1989), pp. 130–164.

    Google Scholar 

  75. S. Kusuoka and D. W. Stroock, “Applications of Malliavin calculus, Part I,” Proc. Taniguchi Intern. Symp. on Stochastic Analysis, Katata-Kyoto, 1982 (ed. by K. Itô), Kinokuniya/North-Holland, 1984, pp. 271–306.

    Google Scholar 

  76. S. Kusuoka and D. W. Stroock, Applications of Malliavin calculus, Part II, J. Fac. Sci. Univ. Tokyo, Sect. IA, Math., 32 (1985), 1–76; Part III, 34 (1987), 391–442.

    MathSciNet  MATH  Google Scholar 

  77. S. Kusuoka and D. W. Stroock, A preprint on the generalized Malliavin calculus.

    Google Scholar 

  78. P. Krée, Introduction aux theories des distributions en dimension infinie, Bull. Soc. Math., France, 46 (1976), 143–162.

    MathSciNet  MATH  Google Scholar 

  79. P. Krée, “Dimension free stochastic calculus in the distribution sense,” Stochastic Analysis, path integration and dynamics, (ed. by K. D. Elworthy and J. C. Zambrini), Pitman Research Notes in Math. Series, 200 (1989), pp. 94–129.

    Google Scholar 

  80. L. D. Landau and E. M. Lifschitz, “Quantum mechanics-nonrelativistic theory,” 2nd ed., Pergamon, 1965.

    Google Scholar 

  81. R. Léandre, Majoration en temps petit de la densité d'une diffusion dégénérée, J. Funct. Analy., 74 (1987), 399–414.

    CrossRef  MATH  Google Scholar 

  82. R. Léandre, “Applications quantitatives et geometriques du calcul de Malliavin,” Stochastic Analysis, Proc. Japanese-French Seminar, 1987, (ed. by M. Métivier and S. Watanabe), Lect. Notes in Math., 1322 (1988), Springer-Verlag, pp. 109–133.

    Google Scholar 

  83. R. Léandre, Development asymptotique de la densite d'une diffusion degeneree, J. Funct. Anal., (to appear).

    Google Scholar 

  84. P. Lévy, Le mouvement brownien plan, Amer. J. Math., 62 (1940), 487–550.

    CrossRef  MathSciNet  MATH  Google Scholar 

  85. P. L. Malliavin, Formule de la moyenne, calcul de perturbations et théorèmes d'annulation pour les formes harmoniques, J. Funct. Anal., 17 (1974), 274–291.

    CrossRef  MathSciNet  MATH  Google Scholar 

  86. P. Malliavin, “Stochastic calculus of variation and hypoelliptic operators,” Proc. Intern. Symp. SDE Kyoto, 1976 (ed. by K. Itô), Kinokuniya/Wiley, pp. 195–263.

    Google Scholar 

  87. P. Malliavin, “C k-hypoellipticity with degeneracy,” Stochastic Analysis, (ed. by A. Friedman and M. Pinsky), Academic Press, 1978, pp. 199–214, 327–340.

    Google Scholar 

  88. P. Malliavin, Sur certain integrales stochastiques oscillantes, C. R. Acad. Sci. Paris, 295 (1982), 295–300.

    MathSciNet  MATH  Google Scholar 

  89. P. Malliavin, “Analyse différentielle sur l'espace de Wiener,” Proc. ICM Warszawa, 1982, Vol. 2, PWN, 1984, pp. 1089–1096.

    Google Scholar 

  90. P. Malliavin, “Implicit functions in finite corank on the Wiener space,” Proc. Taniguchi Intern. Symp. on Stochastic Analysis, Katata-Kyoto, 1982, (ed. by K. Itô), Kinokuniya/North-Holland, 1984, pp. 369–386.

    Google Scholar 

  91. P. Malliavin, Integrals stochastiques oscillants et une formule de Feynman-Kac positive, C. R. Acad. Sci. Paris, 300 (1985), 141–143.

    MathSciNet  MATH  Google Scholar 

  92. P. Malliavin, Analyticité réelle des lois conditionnelles additives, Application à l'état fondamental de l'equation de Schrödinger avec magnétisme,, C. R. Acad. Sci. Paris, 302 (1986), 73–78.

    MathSciNet  Google Scholar 

  93. S. Manabe and I. Shigekawa, “A comparison theorem for the eigenvalue for the covariant Laplacian,” to appear.

    Google Scholar 

  94. H. P. McKean, An upper bound to the spectrum of Δ on a manifold of negative curvature, J. Diff. Geom., 4 (1970), 359–366.

    MathSciNet  MATH  Google Scholar 

  95. H. P. McKean, “Stochastic integrals,” Academic Press, 1972.

    Google Scholar 

  96. H. P. McKean and I. M. Singer, Curvature and the eigenvalues of the Laplacian, J. Diff. Geom., 1 (1967), 43–69.

    MathSciNet  MATH  Google Scholar 

  97. H. Matsumoto, The short time asymptotics of the traces of the heat kernel for the magnetic Schrödinger operators, preprint.

    Google Scholar 

  98. R. B. Melrose and M. E. Taylor, Near peak scattering and the corrected Kirchhoff approximation for a convex obstacle, Adv. in Math., 55 (1985), 242–315.

    CrossRef  MathSciNet  MATH  Google Scholar 

  99. P. A. Meyer, “Qulques results analytiques sur le semigroupe d'Ornstein-Uhlenbeck en dimension infinie,” Theory and applications of random fields, Proc. IFIP-WG/71 Working Conference, (ed. by G. Kallianpur), Lect. Notes in Contr. Inform., 49 (1983), Springer-Verlag, pp. 201–214.

    Google Scholar 

  100. J. Milnor, “Morse theory,” Annals of Math. Studies, 51, Princeton Univ. Press, 1963.

    Google Scholar 

  101. S. Minakshisundaram and A. Pleijel, Some properties of the eigenfunctions of the Laplace operators on Riemannian manifolds, Can. Jour. Math., 1 (1949), 242–256.

    CrossRef  MathSciNet  MATH  Google Scholar 

  102. I. Mitoma, Tightness of probabilities on C([0,1];S') and D([0,1];S'), Ann. Probab., 11 (1983), 989–999.

    CrossRef  MathSciNet  MATH  Google Scholar 

  103. S. A. Molchanov, Diffusion processes and Riemannian geometry, Russian Math., Survey, 30 (1975), 1–63.

    CrossRef  MathSciNet  MATH  Google Scholar 

  104. E. Nelson, unpublished.

    Google Scholar 

  105. K. Odencrantz, The effects of magnetic field on asymptotics of the trace of the heat kernel, J. Funct. Anal., 79 (1988), 398–422.

    CrossRef  MathSciNet  MATH  Google Scholar 

  106. R. E. A. C. Paley and N. Wiener, “Fourier transforms in the complex domain,” Amer. Math. Soc. Coll. Publ., 19, 1934.

    Google Scholar 

  107. V. K. Patodi, Curvature and eigenforms of the Laplace operator, J. Diff. Geom., 5 (1971), 233–249.

    MathSciNet  MATH  Google Scholar 

  108. D. B. Ray, On spectra of second order differential operators, Trans. Am. Math. Soc., 77 (1954), 299–321.

    CrossRef  MathSciNet  MATH  Google Scholar 

  109. J. Rezende, The method of stationary phase for oscillatory integrals on Hilbert space, Comm. Math. Phys., 101 (1985), 187–206.

    CrossRef  MathSciNet  MATH  Google Scholar 

  110. G. de Rham, “Variétés différentiables,” Hermann, Paris, 1960.

    Google Scholar 

  111. M. Riesz, L'intégrale de Riemann-Liouville et le problème de Cauchy, Acta Math., 81 (1949), 1–223.

    CrossRef  MathSciNet  MATH  Google Scholar 

  112. M. Schilder, Some asymptotic formulas for Wiener integrals, Trans. Amer. Math., 125 (1966), 63–85.

    CrossRef  MathSciNet  MATH  Google Scholar 

  113. T. W. Sheu, Spectral properties of differential operators related to stochastic oscillatory integrals, preprint.

    Google Scholar 

  114. I. Shigekawa, Derivatives of Wiener functionals and absolute continuity of induced measures, J. Math. Kyoto Univ., 20 (1980), 263–289.

    MathSciNet  MATH  Google Scholar 

  115. I. Shigekawa, Eigenvalue problems for the Schrödinger operator with the magnetic field on a compact Riemannian manifold, J. Funct. Analy., 75 (1987), 92–127.

    CrossRef  MathSciNet  MATH  Google Scholar 

  116. I. Shigekawa and N. Ueki, A stochastic approach to the Riemann-Roch theorem, Osaka J. Math., 25 (1988), 717–725.

    MathSciNet  MATH  Google Scholar 

  117. I. Shigekawa, N. Ueki and S. Watanabe, A probabilistic proof of the Gauss-Bonnet-Chern theorem for manifold with boundary, Osaka J. Math., 26 (1989) (to appear).

    Google Scholar 

  118. B. Simon, “Functional integration and quantum physics,” Academic Press, 1979.

    Google Scholar 

  119. D. W. Stroock, “Some applications of stochastic calculus to partial differential equations,” Ecole d'Eté de probabilités de Saint-Flour XI-1981, (ed. par P. L. Hennequin), Lect. Notes in Math., 976 (1983), Springer-Verlag, pp. 267–382.

    Google Scholar 

  120. H. Sugita, On a characterization of the Sobolev spaces over an abstract Wiener space, J. Math. Kyoto Univ., 25 (1985), 31–48.

    MathSciNet  MATH  Google Scholar 

  121. H. Sugita, Positive generalized Wiener functionals and potential theory over abstract Wiener space, Osaka J. Math., 25 (1988), 665–696.

    MathSciNet  MATH  Google Scholar 

  122. T. Sunada, “Fundamental groups and Laplacians,” Proc. 21st Intern. Taniguchi Symp. Katata-Kyoto, 1987 (ed. by T. Sunada), Lect. Notes in Math., 1339 (1988), Springer-Verlag.

    Google Scholar 

  123. Y. Takahashi and S. Watanabe, “The probability functionals (Onsager-Machlup functions) of diffusion processes,” Stochastic integrals (ed. by D. Williams), Lect. Notes in Math., 851 (1981), Springer-Verlag, pp. 433–463.

    Google Scholar 

  124. S. Takanobu, Diagonal short time asymptotics of heat kernels for certain degenerate second order differential operators of Hörmander type, Publ. RIMS Kyoto Univ., 24 (1988), 169–203.

    CrossRef  MathSciNet  MATH  Google Scholar 

  125. H. Tamura, Asymptotic distribution of eigenvalues for Schrödinger operators with magnetic fields, Nagoya Math. J., 105 (1987), 49–69.

    CrossRef  MathSciNet  MATH  Google Scholar 

  126. S. Taniguchi, Malliavin's stochastic calculus of variations for manifold-valued Wiener functionals and its applications, Z. Wahrsch. verw. Gebiete, 65 (1983), 269–290.

    CrossRef  MathSciNet  MATH  Google Scholar 

  127. N. Ueki, A probabilistic approach to the Poincare-Hopf theorem, Probab. Th. Rel., Fields, 82 (1989), 271–293.

    CrossRef  MathSciNet  MATH  Google Scholar 

  128. H. Uemura, On a short time expansion of the fundamental solution of heat equations by the method of Wiener functionals, J. Math. Kyoto Univ., 27 (1987), 417–431.

    MathSciNet  MATH  Google Scholar 

  129. H. Uemura and S. Watanabe, “Diffusion processes and heat kernels on certain nilpotent groups,” Stochastic Analysis, Proc. Japanese-French Seminar, 1987, Lect. Notes in Math., 1322 (1987), Springer-Verlag, pp. 37–49.

    Google Scholar 

  130. S. R. S. Varadhan, On the behavior of the fundamental solution of the heat equation with variable coefficients, Comm. Pure Appl. Math., 20 (1967), 431–455.

    CrossRef  MathSciNet  MATH  Google Scholar 

  131. S. R. S. Varadhan, “Large deviations and applications,” CBMS-NSF Regional Conf. Series in Appl. Math., 46, 1984.

    Google Scholar 

  132. S. Watanabe, “Lectures on stochastic differential equations and Malliavin calculus,” Tata Institute of Fundamental Research, Springer-Verlag, 1984.

    Google Scholar 

  133. S. Watanabe, Analysis of Wiener functions (Malliavin calculus) and its applications to heat kernels, Ann. Probab., 15 (1987), 1–39.

    CrossRef  MathSciNet  MATH  Google Scholar 

  134. S. Watanabe, “Generalized Wiener functionals and their applications,” Probability theory and mathematical statistics, Proc. Fifth Japan-USSR Symp. on Probab., Kyoto, 1986 (ed. by S. Watanabe and Y. V. Prohorov), Lect. Notes in Math., 1299 (1988), Springer-Verlag, pp. 541–548.

    Google Scholar 

  135. S. Watanabe, Short time asymptotic problems in Wiener functional integration theory. Applications to heat kernels and index theorems, preprint.

    Google Scholar 

  136. S. Watanabe, Disintegration problems in Wiener functional integrations, preprint.

    Google Scholar 

  137. P. Witten, Supersymmetry and Morse theory, J. Diff. Geom., 17 (1982), 661–692.

    MathSciNet  MATH  Google Scholar 

  138. M. Yor, “Remarques sur une formule de Paul Lévy,” Seminaire de Prob., XIV, 1978/79, (ed. par J. Azema et M. Yor), Lect. Notes in Math., 784, SpringerVerlag, 1980, pp. 343–346.

    Google Scholar 

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© 1990 Springer-Verlag

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Ikeda, N. (1990). Probabilistic methods in the study of asymptotics. In: Hennequin, PL. (eds) École d'Été de Probabilités de Saint-Flour XVIII - 1988. Lecture Notes in Mathematics, vol 1427. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0103043

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  • DOI: https://doi.org/10.1007/BFb0103043

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