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Second term of the logarithmic asymptotics of path integrals. (English) Zbl 0558.28009

Translation from Itogi Nauki Tekh., Ser. Teor. Veroyatn. Mat. Stat. Teor. Kibern. 19, 127-154 (Russian) (1982; Zbl 0531.28014).

MSC:

28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
35K10 Second-order parabolic equations
60B05 Probability measures on topological spaces
60J99 Markov processes

Citations:

Zbl 0531.28014
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References:

[1] V. I. Arnol’d, ?Normal forms of functions in a neighborhood of degenerate critical points,? Usp. Mat. Nauk,29, No. 2, 11?49 (1974).
[2] V. I. Arnol’d, ?Critical points of smooth functions and their normal forms,? Usp. Mat. Nauk,30, No. 5, 3?65 (1975).
[3] I. N. Bernshtein and S. I. Gel’fand, ?Meromorphicity of the function P?,? Funkts. Anal. Prilozhen.,3, No. 1, 84?85 (1969).
[4] A. A. Borovkov, ?Boundary values for random walks and large deviations in function spaces,? Teor. Veroyatn. Primen.,12, No. 4, 635?654 (1967). · Zbl 0178.20004
[5] V. S. Buslaev, ?Path integrals and the asymptotics of solutions of parabolic equations as t?0. Applications to diffraction,? in: Problemy Matematicheskoi Fiziki, No. 2, Leningrad State Univ. (1967), pp. 85?107. · Zbl 0164.12602
[6] A. N. Varchenko, ?Newton polygons and estimates of oscillating integrals,? Funks. Anal. Prilozhen.,10, No. 3, 13?38 (1976).
[7] A. D. Venttsel’ and M. I. Freidlin, Fluctuations in Dynamical Systems under Small Random Perturbations [in Russian], Nauka, Moscow (1979).
[8] Yu. Gertner, ?Theorems on large deviations for a class of stochastic processes,? Teor. Veroyatn. Primen.,21, No. 1, 95?106 (1976).
[9] Yu. Gertner, ?On large deviations from an invariant measure,? Teor. Veroyatn. Primen.,22, No. 1, 27?42 (1977).
[10] A. G. Grin’, ?On small random momentum perturbations of dynamical systems,? Teor. Veroyatn. Primen.,20, No. 1, 150?158 (1975).
[11] V. N. Dubrovskii, ?Precise asymptotic formulas of Laplace type for Markov processes,? Dokl. Akad. Nauk SSSR,226, No. 5, 1001?1004 (1976). · Zbl 0369.60030
[12] Yu. I. Kifer, ?On the asymptotics of transition densities of processes with small diffusion,? Teor. Veroyatn. Primen.,21, No. 3, 527?536 (1976). · Zbl 0367.60035
[13] V. P. Maslov, Perturbation Theory and Asymptotic Methods [in Russian], Moscow State Univ. (1965). · Zbl 0653.35002
[14] V. P. Maslov, ?Global asymptotic expansion of the probability of large deviations and its relation to the quasiclassical expansion,? Report at a Session of the Moscow Mathematical Society, February 20, 1979, Usp. Mat. Nauk,34, No. 5, 213 (1979).
[15] V. P. Maslov and M. V. Fedoryuk, Quasiclassical Approximation for the Equations of Quantum Mechanics [in Russian], Nauka, Moscow (1976). · Zbl 0449.58002
[16] V. P. Maslov and M. V. Fedoryuk, ?On the second term of the logarithmic asymptotics of Laplace integrals,? Mat. Zametki,30, No. 5 (1981). · Zbl 0489.60031
[17] J. W. Milnor, Morse Theory, Princeton Univ. Press (1963).
[18] A. A. Mogul’skiii, ?Large deviations for trajectories of multidimensional random walks,? Teor. Veroyatn. Primen.,21, No. 2, 309?323 (1976).
[19] S. A. Molchanov, ?Diffusion processes and Riemannian geometry,? Usp. Mat. Nauk,30, No. 1, 3?59 (1975).
[20] V. V. Petrov, Sums of Independent Random Variables, Springer-Verlag (1975). · Zbl 0322.60043
[21] R. T. Rockefeller, Convex Analysis, Princeton Univ. Press (1970).
[22] M. V. Fedoryuk, The Method of Descent [in Russian], Nauka, Moscow (1977). · Zbl 0463.41020
[23] M. I. Freidlin, ?On the stability of high-reliability systems,? Teor. Veroyatn. Primen.,20, No. 3, 584?595 (1975).
[24] M. I. Freidlin, ?The principle of averaging and theorems on large deviations,? Usp. Mat, Nauk,33, No. 5, 107?160 (1978).
[25] P. Hartman, Ordinary Differential Equations, S. M. Hartman (1973).
[26] S. A. Albeverio and R. J. Höegh-Krohn, ?Oscillatory integrals and the method of stationary phase in infinitely many dimensions with applications to the classical limit of quantum mechanics. I,? Invent. Math.,40, No. 1, 59?106 (1977). · Zbl 0449.35092
[27] S. A. Albeverio and R. J. Höegh-Krohn, ?Mathematical theory of Feynman path integrals,? Lect. Notes Math., No. 523 (1976).
[28] Ph. Choquard, ?Traitement semiclassique des forces générales dans la representation de Feynman,? Helv. Phys. Acta,28, No. 2?3, 89?157 (1955). · Zbl 0064.21603
[29] G. Dangelmayr and W. Veit, ?Semiclassical approximation of path integrals on and near caustics in terms of catastrophes,? Ann. Phys.,118, No. 1, 108?138 (1979). · Zbl 0425.58008
[30] D. W. McLaughlin, ?Path integrals, asymptotics, and singular perturbations,? J. Math. Phys.,13, No. 5, 784?796 (1972).
[31] L. S. Schulman, ?Caustics and multivaluedness. Two results of adding path amplitudes,? in: Functional Integration and Its Applications, Oxford Univ. Press, London (1975). · Zbl 0336.46065
[32] S. R. S. Varadhan, ?On the behavior of the fundamental solution of the heat equation with variable coefficients,? Commun. Pure Appl. Math.,20, No. 2, 431?455 (1967). · Zbl 0155.16503
[33] S. R. S. Varadhan, ?Diffusion processes in a small time interval,? Commun. Pure Appl. Math.,20, No. 4, 659?685 (1967). · Zbl 0278.60051
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