Rotation of Gaussian processes on function space

  • Seung Jun Chang
  • David Skoug
  • Jae Gil ChoiEmail author
Original Paper


The purpose of this paper is to investigate a more general rotation property of Gaussian processes on the function space \(C_{a,b}[0,T]\). The function space \(C_{a,b}[0,T]\) can be induced by a generalized Brownian motion process. The Gaussian processes used in this paper are neither centered nor stationary.


Generalized Brownian motion process Paley–Wiener–Zygmund stochastic integral Gaussian process Rotation of Gaussian processes 

Mathematics Subject Classification

Primary 46G12 60G15 Secondary 28C20 60J65 



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. The present research was conducted by the research fund of Dankook University in 2019.


  1. 1.
    Bearman, J.E.: Rotations in the product of two Wiener spaces. Proc. Am. Math. Soc. 3, 129–137 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cameron, R.H., Storvick, D.A.: An operator valued Yeh-Wiener integral, and a Wiener integral equation. Indiana Univ. Math. J. 25, 235–258 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chang, S.J., Choi, J.G., Ko, A.Y.: Multiple generalized analytic Fourier–Feynman transform via rotation of Gaussian paths on function space. Banach J. Math. Anal. 9, 58–80 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chang, S.J., Choi, J.G., Skoug, D.: Integration by parts formulas involving generalized Fourier–Feynman transforms on function space. Trans. Am. Math. Soc. 355, 2925–2948 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chang, S.J., Choi, J.G., Skoug, D.: Generalized Fourier–Feynman transforms, convolution products, and first variations on function space. Rocky Mt. J. Math. 40, 761–788 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chang, S.J., Skoug, D.: Generalized Fourier–Feynman transforms and a first variation on function space. Integr. Transf. Spec. Funct. 14, 375–393 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Choi, J.G., Chung, H.S., Chang, S.J.: Sequential generalized transforms on function space. Abst. Appl. Anal. 2013, 565832 (2013)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Choi, J.G., Skoug, D., Chang, S.J.: A multiple generalized Fourier-Feynman transform via a rotation on Wiener space. Int. J. Math. 23, 1250068 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chung, H.S., Choi, J.G., Chang, S.J.: A Fubini theorem on a function space and its applications. Banach J. Math. Anal. 7, 173–185 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Huffman, T., Park, C., Skoug, D.: Analytic Fourier–Feynman transforms and convolution. Trans. Am. Math. Soc. 347, 661–673 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Huffman, T., Park, C., Skoug, D.: Convolutions and Fourier–Feynman transforms of functionals involving multiple integrals. Mich. Math. J. 43, 247–261 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Huffman, T., Park, C., Skoug, D.: Convolution and Fourier–Feynman transforms. Rocky Mt. J. Math. 27, 827–841 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Huffman, T., Park, C., Skoug, D.: Generalized transforms and convolutions. Int. J. Math. Math. Sci. 20, 19–32 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Skoug, D., Storvick, D.: A survey of results involving transforms and convolutions in function space. Rocky Mt. J. Math. 34, 1147–1175 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Yeh, J.: Singularity of Gaussian measures on function spaces induced by Brownian motion processes with non-stationary increments. Ill. J. Math. 15, 37–46 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Yeh, J.: Stochastic Processes and the Wiener Integral. Marcel Dekker Inc, New York (1973)zbMATHGoogle Scholar

Copyright information

© The Royal Academy of Sciences, Madrid 2019

Authors and Affiliations

  1. 1.Department of MathematicsDankook UniversityCheonanKorea
  2. 2.Department of MathematicsUniversity of Nebraska-LincolnLincolnUSA

Personalised recommendations