Bulletin of the Korean Mathematical Society (대한수학회보)

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EVALUATION FORMULA FOR WIENER INTEGRAL OF POLYNOMIALS IN TERMS OF NATURAL DUAL PAIRINGS ON ABSTRACT WIENER SPACES

  • Received : 2021.07.22
  • Accepted : 2022.07.15
  • Published : 2022.09.30
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Abstract

In this paper, we establish an evaluation formula to calculate the Wiener integral of polynomials in terms of natural dual pairings on abstract Wiener spaces (H, B, 𝜈). To do this we first derive a translation theorem for the Wiener integral of functionals associated with operators in 𝓛(B), the Banach space of bounded linear operators from B to itself. We then apply the translation theorem to establish an integration by parts formula for the Wiener integral of functionals combined with operators in 𝓛(B). We finally apply this parts formula to evaluate the Wiener integral of certain polynomials in terms of natural dual pairings.

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Acknowledgement

The authors would like to express their gratitude to the editor and the referees for their valuable comments and suggestions which have improved the original paper.

References

  1. J. C. Baez, I. E. Segal, and Z.-F. Zhou, Introduction to Algebraic and Constructive Quantum Field Theory, Princeton Series in Physics, Princeton University Press, Princeton, NJ, 1992. https://doi.org/10.1515/9781400862504
  2. R. H. Cameron, The first variation of an indefinite Wiener integral, Proc. Amer. Math. Soc. 2 (1951), 914-924. https://doi.org/10.2307/2031708
  3. R. H. Cameron and R. E. Graves, Additive functionals on a space of continuous functions. I, Trans. Amer. Math. Soc. 70 (1951), 160-176. https://doi.org/10.2307/1990530
  4. R. H. Cameron and W. T. Martin, Transformations of Wiener integrals under translations, Ann. of Math. (2) 45 (1944), 386-396. https://doi.org/10.2307/1969276
  5. R. H. Cameron and D. A. Storvick, Feynman integral of variations of functionals, in Gaussian random fields (Nagoya, 1990), 144-157, Ser. Probab. Statist., 1, World Sci. Publ., River Edge, NJ, 1991.
  6. D. M. Chung, Scale-invariant measurability in abstract Wiener spaces, Pacific J. Math. 130 (1987), no. 1, 27-40. http://projecteuclid.org/euclid.pjm/1102690291 102690291
  7. G. B. Folland, Real Analysis, second edition, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1999.
  8. 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, 31-42, Univ. California Press, Berkeley, CA, 1967.
  9. N. Hayek, B. J. Gonzalez, and E. R. Negrin, The second quantization and its general integral finite-dimensional representation, Integral Transforms Spec. Funct. 13 (2002), no. 4, 373-378. https://doi.org/10.1080/10652460213525
  10. N. Hayek, H. M. Srivastava, B. J. Gonzalez, and E. R. Negrin, A family of Wiener transforms associated with a pair of operators on Hilbert space, Integral Transforms Spec. Funct. 24 (2013), no. 1, 1-8. https://doi.org/10.1080/10652469.2011.648379
  11. S. Janson, Gaussian Hilbert spaces, Cambridge Tracts in Mathematics, 129, Cambridge University Press, Cambridge, 1997. https://doi.org/10.1017/CBO9780511526169
  12. G. Kallianpur and C. Bromley, Generalized Feynman integrals using analytic continuation in several complex variables, in Stochastic analysis and applications, 217-267, Adv. Probab. Related Topics, 7, Dekker, New York, 1984.
  13. G. Kallianpur, D. Kannan, and R. L. Karandikar, Analytic and sequential Feynman integrals on abstract Wiener and Hilbert spaces, and a Cameron-Martin formula, Ann. Inst. H. Poincare Probab. Statist. 21 (1985), no. 4, 323-361.
  14. J. Kuelbs, Abstract Wiener spaces and applications to analysis, Pacific J. Math. 31 (1969), 433-450. http://projecteuclid.org/euclid.pjm/1102977879 102977879
  15. H.-H. Kuo, Gaussian measures in Banach spaces, Lecture Notes in Mathematics, Vol. 463, Springer-Verlag, Berlin, 1975.
  16. H.-H. Kuo and Y.-J. Lee, Integration by parts formula and the Stein lemma on abstract Wiener space, Commun. Stoch. Anal. 5 (2011), no. 2, 405-418. https://doi.org/10.31390/cosa.5.2.10
  17. Y. J. Lee, Applications of the Fourier-Wiener transform to differential equations on infinite-dimensional spaces. I, Trans. Amer. Math. Soc. 262 (1980), no. 1, 259-283. https://doi.org/10.2307/1999982
  18. Y. J. Lee, Integral transforms of analytic functions on abstract Wiener spaces, J. Funct. Anal. 47 (1982), no. 2, 153-164. https://doi.org/10.1016/0022-1236(82)90103-3
  19. U. G. Lee and J. G. Choi, An extension of the Cameron-Martin translation theorem via Fourier-Hermite functionals, Arch. Math. (Basel) 115 (2020), no. 6, 679-689. https://doi.org/10.1007/s00013-020-01524-6
  20. E. R. Negrin, Integral representation of the second quantization via the Segal duality transform, J. Funct. Anal. 141 (1996), no. 1, 37-44. https://doi.org/10.1006/jfan.1996.0120
  21. R. E. A. C. Paley, N. Wiener, and A. Zygmund, Notes on random functions, Math. Z. 37 (1933), no. 1, 647-668. https://doi.org/10.1007/BF01474606
  22. C. Park and D. Skoug, A note on Paley-Wiener-Zygmund stochastic integrals, Proc. Amer. Math. Soc. 103 (1988), no. 2, 591-601. https://doi.org/10.2307/2047184
  23. C. Park, D. Skoug, and D. Storvick, Relationships among the first variation, the convolution product and the Fourier-Feynman transform, Rocky Mountain J. Math. 28 (1998), no. 4, 1447-1468. https://doi.org/10.1216/rmjm/1181071725
  24. C. Park, D. Skoug, and D. Storvick, Fourier-Feynman transforms and the first variation, Rend. Circ. Mat. Palermo (2) 47 (1998), no. 2, 277-292. https://doi.org/10.1007/BF02844368
  25. I. E. Segal, Tensor algebras over Hilbert spaces. I, Trans. Amer. Math. Soc. 81 (1956), 106-134. https://doi.org/10.2307/1992855
  26. I. Shigekawa, Derivatives of Wiener functionals and absolute continuity of induced measures, J. Math. Kyoto Univ. 20 (1980), no. 2, 263-289. https://doi.org/10.1215/kjm/ 1250522278