Asymptotic analysis of Gaussian integrals. I. Isolated minimum points
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- by Richard S. Ellis and Jay S. Rosen PDF
- Trans. Amer. Math. Soc. 273 (1982), 447-481 Request permission
Abstract:
This paper derives the asymptotic expansions of a wide class of Gaussian function space integrals under the assumption that the minimum points of the action are isolated. Degenerate as well as nondegenerate minimum points are allowed. This paper also derives limit theorems for related probability measures which correspond roughly to the law of large numbers and the central limit theorem. In the degenerate case, the limits are non-Gaussian.References
- Melvin S. Berger, Nonlinearity and functional analysis, Pure and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1977. Lectures on nonlinear problems in mathematical analysis. MR 0488101 Coleman, The uses of instantons, Lectures delivered at the 1977 International School of Subnuclear Physics, Ettore Majorana.
- M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time. III, Comm. Pure Appl. Math. 29 (1976), no. 4, 389–461. MR 428471, DOI 10.1002/cpa.3160290405
- Richard S. Ellis and Charles M. Newman, The statistics of Curie-Weiss models, J. Statist. Phys. 19 (1978), no. 2, 149–161. MR 503332, DOI 10.1007/BF01012508
- R. S. Ellis and J. S. Rosen, Asymptotics of certain random fields on a circle, Random fields, Vol. I, II (Esztergom, 1979) Colloq. Math. Soc. János Bolyai, vol. 27, North-Holland, Amsterdam-New York, 1981, pp. 279–321. MR 712679
- Richard S. Ellis and Jay S. Rosen, Laplace’s method for Gaussian integrals with an application to statistical mechanics, Ann. Probab. 10 (1982), no. 1, 47–66. MR 637376 Erdélyi, Asymptotics expansions, Dover, New York, 1956.
- M. I. Freĭdlin, An action functional for a certain class of random processes, Teor. Verojatnost. i Primenen. 17 (1972), 536–541 (Russian, with English summary). MR 0307314
- Ĭ. Ī. Gīhman and A. V. Skorohod, The theory of stochastic processes. I, Die Grundlehren der mathematischen Wissenschaften, Band 210, Springer-Verlag, New York-Heidelberg, 1974. Translated from the Russian by S. Kotz. MR 0346882
- I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by A. Feinstein. MR 0246142
- G. Kallianpur and H. Oodaira, Freĭdlin-Wentzell type estimates for abstract Wiener spaces, Sankhyā Ser. A 40 (1978), no. 2, 116–137. MR 546403
- Martin Pincus, Gaussian processes and Hammerstein integral equations, Trans. Amer. Math. Soc. 134 (1968), 193–214. MR 231439, DOI 10.1090/S0002-9947-1968-0231439-1
- Balram S. Rajput, The support of Gaussian measures on Banach spaces, Teor. Verojatnost. i Primenen. 17 (1972), 775–782 (English, with Russian summary). MR 0324737
- Frigyes Riesz and Béla Sz.-Nagy, Functional analysis, Frederick Ungar Publishing Co., New York, 1955. Translated by Leo F. Boron. MR 0071727
- M. Schilder, Some asymptotic formulas for Wiener integrals, Trans. Amer. Math. Soc. 125 (1966), 63–85. MR 201892, DOI 10.1090/S0002-9947-1966-0201892-6
- Lawrence S. Schulman, Techniques and applications of path integration, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1981. MR 601595
- Barry Simon, Functional integration and quantum physics, Pure and Applied Mathematics, vol. 86, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR 544188
- A. V. Skorohod, Integration in Hilbert space, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 79, Springer-Verlag, New York-Heidelberg, 1974. Translated from the Russian by Kenneth Wickwire. MR 0466482
- N. M. Temme (ed.), Nonlinear analysis. Vol. 1, MC Syllabus, No. 26.1, Mathematisch Centrum, Amsterdam, 1976. With contributions by L. A. Peletier, T. H. Koornwinder, N. M. Temme, O. Diekmann and I. G. Sprinkhuizen-Kuyper. MR 0493535
- S. R. S. Varadhan, Asymptotic probabilities and differential equations, Comm. Pure Appl. Math. 19 (1966), 261–286. MR 203230, DOI 10.1002/cpa.3160190303 D. Wentzell, Theorems on the action functional for Gaussian random functions, Theor. Probab. Appl. 17 (1972), 515-517.
- F. W. Wiegel, Path integral methods in statistical mechanics, Phys. Rep. 16C (1975), 57–114. MR 424132, DOI 10.1016/0370-1573(75)90030-7
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 273 (1982), 447-481
- MSC: Primary 60G15; Secondary 28C20, 58D20, 81C35
- DOI: https://doi.org/10.1090/S0002-9947-1982-0667156-0
- MathSciNet review: 667156