Taiwanese Journal of Mathematics

An Analytic Version of Wiener-Itô Decomposition on Abstract Wiener Spaces

Yuh-Jia Lee and Hsin-Hung Shih

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Abstract

In this paper, we first establish an analogue of Wiener-Itô theorem on finite-dimensional Gaussian spaces through the inverse $S$-transform, that is, the Gauss transform on Segal-Bargmann spaces. Based on this point of view, on infinite-dimensional abstract Wiener space $(H,B)$, we apply the analyticity of the $S$-transform, which is an isometry from the $L^2$-space onto the Bargmann-Segal-Dwyer space, to study the regularity. Then, by defining the Gauss transform on Bargmann-Segal-Dwyer space and showing the relationship with the $S$-transform, an analytic version of Wiener-Itô decomposition will be obtained.

Article information

Source
Taiwanese J. Math., Volume 23, Number 2 (2019), 453-471.

Dates
Received: 8 May 2018
Accepted: 10 December 2018
First available in Project Euclid: 28 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1545966024

Digital Object Identifier
doi:10.11650/tjm/181207

Mathematical Reviews number (MathSciNet)
MR3936008

Zentralblatt MATH identifier
07055577

Subjects
Primary: 60B11: Probability theory on linear topological spaces [See also 28C20] 46E20: Hilbert spaces of continuous, differentiable or analytic functions 46E50: Spaces of differentiable or holomorphic functions on infinite- dimensional spaces [See also 46G20, 46G25, 47H60]
Secondary: 28C20: Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11]

Keywords
abstract Wiener space abstract Wiener measure Wiener-Itô decomposition Segal-Bargmann transform Gauss transform

Citation

Lee, Yuh-Jia; Shih, Hsin-Hung. An Analytic Version of Wiener-Itô Decomposition on Abstract Wiener Spaces. Taiwanese J. Math. 23 (2019), no. 2, 453--471. doi:10.11650/tjm/181207. https://projecteuclid.org/euclid.twjm/1545966024


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References

  • V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, Comm. Pure Appl. Math. 14 (1961), 187–214.
  • S. Bochner and W. T. Martin, Several Complex Variables, Princeton Mathematical Series 10, Princeton University Press, Princeton, N.J., 1948.
  • I. M. Gel'fand and N. Y. Vilenkin, Generalized Functions, Vol. 4: Applications of Harmonic Analysis, Academic Press, New York, 1964.
  • L. Gross, Abstract Wiener spaces, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), Vol. II: Contributions to Probability Theory, Part 1, (1965), 31–42.
  • ––––, Potential theory on Hilbert space, J. Functional Analysis 1 (1967), 123–181.
  • B. C. Hall, Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, Springer, New York, 2013.
  • E. Hille and R. S. Phillips, Functional Analysis and Semi-groups, American Mathematical Society Colloquium Publications 31, American Mathematical Society, Providence, R.I., 1957.
  • K. Itô, Multiple Wiener integral, J. Math. Soc. Japan 3 (1951), 157–169.
  • H.-H. Kuo, Gaussian Measures in Banach Spaces, Lecture Notes in Mathematics 463, Springer-Verlag, Berlin, 1975.
  • ––––, White Noise Distribution Theory, Probability and Stochastics Series, CRC Press, Boca Raton, FL, 1996.
  • Y.-J. Lee, Sharp inequalities and regularity of heat semigroup on infinite-dimensional spaces, J. Funct. Anal. 71 (1987), no. 1, 69–87.
  • ––––, On the convergence of Wiener-Itô decomposition, Bull. Inst. Math. Acad. Sinica 17 (1989), no. 4, 305–312.
  • ––––, Analytic version of test functionals, Fourier transform, and a characterization of measures in white noise calculus, J. Funct. Anal. 100 (1991), no. 2, 359–380.
  • Y.-J. Lee and H.-H. Shih, The Clark formula of generalized Wiener functionals, in: Quantum Information IV, (Nagoya, 2001), 127–145, World Sci. Publ., River Edge, NJ, 2002.
  • N. Obata, White Noise Calculus and Fock Space, Lecture Notes in Mathematics 1577, Springer-Verlag, Berlin, 1994.
  • R. Schatten, Norm Ideals of Completely Continuous Operators, Ergebnisse der Mathematik und ihrer Grenzgebiete. N. F., Heft 27 Springer-Verlag, Berlin, 1960.
  • W. Schoutens, Stochastic Processes and Orthogonal Polynomials, Lecture Notes in Statistics 146, Springer-Verlag, New York, 2000.
  • R. L. Wheeden and A. Zygmund, Measure and Integral: An Introduction to Real Analysis, Pure and Applied Mathematics 43, Marcel Dekker, New York, 1977.
  • N. Wiener, The homogeneous chaos, Amer. J. Math. 60 (1938), no. 4, 897–936.