Infinite dimensional oscillatory integrals with polynomial phase function and the trace formula for the heat semigroup
From probability to geometry (I) - Volume in honor of the 60th birthday of Jean-Michel Bismut, Astérisque no. 327  (2009), p. 17-45
@incollection{AST_2009__327__17_0,
     author = {Albeverio, Sergio and Mazzucchi, Sonia},
     title = {Infinite dimensional oscillatory integrals with polynomial phase function and the trace formula for the heat semigroup},
     booktitle = {From probability to geometry (I) - Volume in honor of the 60th birthday of Jean-Michel Bismut},
     editor = {Dai Xianzhe and L\'eandre R\'emi and Xiaonan Ma and Zhang Weiping},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {327},
     year = {2009},
     pages = {17-45},
     zbl = {1208.28009},
     mrnumber = {2642350},
     language = {en},
     url = {http://www.numdam.org/item/AST_2009__327__17_0}
}
Albeverio, Sergio; Mazzucchi, Sonia. Infinite dimensional oscillatory integrals with polynomial phase function and the trace formula for the heat semigroup, in From probability to geometry (I) - Volume in honor of the 60th birthday of Jean-Michel Bismut, Astérisque, no. 327 (2009), pp. 17-45. http://www.numdam.org/item/AST_2009__327__17_0/

[1] S. Albeverio - "Wiener and Feynman path integrals and their applications", in Proceedings of the Norbert Wiener Centenary Congress (East Lansing, MI, 1994), 1994. | Zbl 0899.60058

[2] S. Albeverio & T. Arede - "The relation between quantum mechanics and classical mechanics: a survey of some mathematical aspects", in Chaotic Behavior in Quantum Systems, Theory and Applications (G. Casati et al., eds.), Plenum, 1985. | Article

[3] S. Albeverio, T. Arede & M. D. Faria - "Remarks on nonlinear filtering problems: white noise representation and asymptotic expansions", in Stochastic processes, physics and geometry (Ascona and Locarno, 1988), World Sci. Publ., Teaneck, NJ, 1990, p. 77-86. | MR 1124203

[4] S. Albeverio, P. Blanchard & R. Høegh-Krohn - "Feynman path integrals and the trace formula for the Schrödinger operators", Comm. Math. Phys. 83 (1982), p. 49-76. | Article | MR 648358 | Zbl 0493.35039

[5] S. Albeverio & Z. Brzeźniak - "Finite-dimensional approximation approach to oscillatory integrals and stationary phase in infinite dimensions", J. Funct. Anal. 113 (1993), p. 177-244. | Article | MR 1214902 | Zbl 0779.46040

[6] S. Albeverio & Z. Brzezniak,"Feynman path integrals as infinite-dimensional oscillatory integrals: some new developments", Acta Appl. Math. 35 (1994), p. 5-26. | Article | MR 1286497 | Zbl 0813.58009

[7] S. Albeverio, R. Høegh-Krohn & S. Mazzucchi - "Mathematical theory of Feynman path integrals-An introduction", second corrected and enlarged edition, Lecture Notes in Math., vol. 523, Springer, 2008. | MR 2453734 | Zbl 1222.46001

[8] S. Albeverio & S. Mazzucchi - "Feynman path integrals for polynomially growing potentials", J. Funct Anal. 221 (2005), p. 83-121. | Article | MR 2124898 | Zbl 1095.81041

[9] S. Albeverio & S. Mazzucchi, "Generalized Fresnel integrals", Bull. Sci. Math. 129 (2005), p. 1-23. | Article | MR 2114624 | Zbl 1096.28010

[10] S. Albeverio & I. Mitoma - "Asymptotic expansion of perturbative Chern-Simons theory via Wiener space", Bull. Sci. Math. 133 (2009), p. 272-314. | Article | MR 2512830 | Zbl 1169.57030

[11] S. Albeverio, A. Boutet De Monvel-Berthier & Z. Brzeźniak - "Stationary phase method in infinite dimensions by finite-dimensional approximations: applications to the Schrödinger equation", Potential Anal. 4 (1995), p. 469-502. | Article | MR 1354807 | Zbl 0845.58019

[12] S. Albeverio, A. Boutet De Monvel-Berthier & Z. Rzeźniak, "The trace formula for Schrödinger operators from infinite-dimensional oscillatory integrals", Math. Nachr. 182 (1996), p. 21-65. | Article | MR 1419888 | Zbl 0866.58020

[13] S. Albeverio, H. Röckle & V. Steblovskaya - "Asymptotic expansions for Ornstein-Uhlenbeck semigroups perturbed by potentials over Banach spaces", Stochastics Stochastics Rep. 69 (2000), p. 195-238. | Article | MR 1760977 | Zbl 0973.60064

[14] S. Albeverio & V. Steblovskaya - "Asymptotics of infinite-dimensional integrals with respect to smooth measures. I", Infin. Dimens. Anal. Quantum Probab. Relat. Top. 2 (1999), p. 529-556. | Article | MR 1810812 | Zbl 1043.46507

[15] R. Azencott & H. Doss - "L'équation de Schrödinger quand h tend vers zéro: une approche probabiliste", in Stochastic aspects of classical and quantum systems (Marseille, 1983), Lecture Notes in Math., vol. 1109, Springer, 1985, p. 1-17. | Article | MR 805986 | Zbl 0555.60040

[16] G. Ben Arous - "Methods de Laplace et de la phase stationnaire sur l'espace de Wiener", Stochastics 25 (1988), p. 125-153. | Article | MR 999365 | Zbl 0666.60026

[17] G. Ben Arous & R. Léandre - "Décroissance exponentielle du noyau de la chaleur sur la diagonale. II", Probab. Theory Related Fields 90 (1991), p. 377-402. | Article | MR 1133372 | Zbl 0734.60027

[18] J.-M. Bismut - Large deviations and the Malliavin calculus, Progress in Math., vol. 45, Birkhäuser, 1984. | MR 755001 | Zbl 0537.35003

[19] R. H. Cameron - "A family of integrals serving to connect the Wiener and Feynman integrals", J. Math. and Phys. 39 (1960/1961), p. 126-140. | Article | MR 127776 | Zbl 0096.06901

[20] Y. Colin De Verdière - "Singular Lagrangian manifolds and semiclassical analysis", Duke Math. J. 116 (2003), p. 263-298. | Article | MR 1953293 | Zbl 1074.53066

[21] R. S. Ellis & J. S. Rosen - "Asymptotic analysis of Gaussian integrals. II. Manifold of minimum points", Comm. Math. Phys. 82 (1981/82), p. 153-181. | Article | MR 639055 | Zbl 0532.28013

[22] R. S. Ellis & J. S. Rosen, "Asymptotic analysis of Gaussian integrals. I. Isolated minimum points", Trans. Amer. Math. Soc. 273 (1982), p. 447-481. | Article | MR 667156 | Zbl 0521.28009

[23] D. Elworthy & A. Truman - "Feynman maps, Cameron-Martin formulae and an-harmonic oscillators", Ann. Inst. H. Poincaré Phys. Théor. 41 (1984), p. 115-142. | Numdam | MR 769152 | Zbl 0578.28013

[24] L. Gross - "Abstract Wiener spaces", in Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif, 1965/66), Vol. II: Contributions to Probability Theory, Part 1, Univ. California Press, 1967, p. 31-42. | MR 212152 | Zbl 0187.40903

[25] M. C. Gutzwiller - Chaos in classical and quantum mechanics, Interdisciplinary Applied Mathematics, vol. 1, Springer, 1990. | MR 1077246 | Zbl 0727.70029

[26] T. Hida, H. H. Kuo, J. Potthoff & L. Streit - White noise, Mathematics and its Applications, vol. 253, Kluwer Academic Publishers Group, 1993. | MR 1244577 | Zbl 0771.60048

[27] L. Hörmander - The analysis of linear partial differential operators. I, Grund. Math. Wiss., vol. 256, Springer, 1983. | MR 717035 | Zbl 0521.35001

[28] G. W. Johnson & M. L. Lapidus - The Feynman integral and Feynman's operational calculus, Oxford Mathematical Monographs, The Clarendon Press Oxford Univ. Press, 2000. | MR 1771173 | Zbl 0952.46044

[29] G. Kallianpur, D. Kannan & R. L. Karandikar - "Analytic and sequential Feynman integrals on abstract Wiener and Hilbert spaces, and a Cameron-Martin formula", Ann. Inst. H. Poincaré Probab. Statist. 21 (1985), p. 323-361. | Numdam | MR 823080 | Zbl 0583.60049

[30] G. Kallianpur & H. Oodaira - "Freĭdlin-Wentzell type estimates for abstract Wiener spaces", Sankhyā Ser. A 40 (1978), p. 116-137. | MR 546403 | Zbl 0419.60035

[31] V. N. Kolokoltsov - Semiclassical analysis for diffusions and stochastic processes, Lecture Notes in Math., vol. 1724, Springer, 2000. | MR 1755149 | Zbl 0951.60001

[32] H. H. Kuo - Gaussian measures in Banach spaces, Lecture Notes in Math., vol. 463, Springer, 1975. | MR 461643 | Zbl 0306.28010

[33] S. Lang - Complex analysis, fourth ed., Graduate Texts in Math., vol. 103, Springer, 1999. | MR 1659317 | Zbl 0933.30001

[34] R. Léandre - "Applications quantitatives et géométriques du calcul de Malliavin", in Stochastic analysis (Paris, 1987), Lecture Notes in Math., vol. 1322, Springer, 1988, p. 109-133. | Article | MR 962867 | Zbl 0666.60014

[35] V. P. Maslov - Théorie des perturbations et méthodes asymptotiques, Dunod, 1972. | Zbl 0247.47010

[36] S. Mazzucchi - Mathematical Feynman path integrals and their applications, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2009. | Article | MR 2537928 | Zbl 1173.81002

[37] E. Nelson - "Feynman integrals and the Schrödinger equation", J. Mathematical Phys. 5 (1964), p. 332-343. | Article | MR 161189 | Zbl 0133.22905

[38] D. Nualart & V. Steblovskaya - "Asymptotics of oscillatory integrals with quadratic phase function on Wiener space", Stochastics Stochastics Rep. 66 (1999), p. 293-309. | Article | MR 1692864 | Zbl 0931.60042

[39] M. Pincus - "Gaussian processes and Hammerstein integral equations", Trans. Amer. Math. Soc. 134 (1968), p. 193-214. | Article | MR 231439 | Zbl 0175.12502

[40] V. I. Piterbarg - Asymptotic methods in the theory of Gaussian processes and fields, Translations of Mathematical Monographs, vol. 148, Amer. Math. Soc., 1996. | MR 1361884 | Zbl 0841.60024

[41] V. I. Piterbarg &V. R. Fatalov - "The Laplace method for probability measures in Banach spaces", Uspekhi Mat. Nauk 50 (1995), p. 57-150. | MR 1379077 | Zbl 0871.46020

[42] M. Reed & B. Simon - Methods of modem mathematical physics. II. Fourier analysis, self-adjointness, Academic Press, 1975. | MR 493420

[43] S. Rossignol - "Développements asymptotiques d'intégrales de Laplace sur l'espace de Wiener dans le cas dégénéré", C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), p. 971-974. | MR 1249371 | Zbl 0791.60051

[44] M. Schilder - "Some asymptotic formulas for Wiener integrals", Trans. Amer. Math. Soc. 125 (1966), p. 63-85. | Article | MR 201892 | Zbl 0156.37602

[45] B. Simon - Trace ideals and their applications, London Mathematical Society Lecture Note Series, vol. 35, Cambridge Univ. Press, 1979. | MR 541149 | Zbl 0423.47001

[46] B. Simon, Functional integration and quantum physics, second ed., AMS Chelsea Publishing, Providence, RI, 2005. | MR 2105995 | Zbl 1061.28010

[47] E. C. Tichmarsch - The theory of functions, Oxford Univ. Press, 1939. | MR 3728294

[48] T. J. Zastawniak - "Equivalence of Albeverio-Høegh-Krohn-Feynman integral for an-harmonic oscillators and the analytic Feynman integral", Univ. Iagel. Acta Math. 28 (1991), p. 187-199. | MR 1136792 | Zbl 0762.46034