Abstract
Based on the complex hyperbolic geometry associated with discrete series of SU(1, 1), we construct a quasi-invariant and ergodic measure on infinite product of Poincaré disc and a hyperbolic analogue of numerical Wiener space which turns out to be a nonlinear deformation of the Wiener space. An integration by parts formula is established. We also investigate the orthogonal decomposition of the L 2-holomorphic functions which is an analogue of the Wiener–Itô–Segal decomposition. In the zero-curvature and large spin limit, we recover the linear Wiener space.
Similar content being viewed by others
REFERENCES
Bargmann, V. (1961). Remarks on a Hilbert space of analytic functions. Proc. Nat. Acad. Sci. U.S.A. 48, 199–204.
Bargmann, V. (1947). Irreducible unitary representation of the Lorentz group. Ann. Math. 48, 568–640.
Gross, L. (1967). Potential theory on a Hilbert space. J. Funct. Anal. 1, 123–181.
Itô, K. (1953). Complex multiple Wiener integrals. Japan J. Math. 22, 63–86.
Itô, K. (1984). Foundations of Stochastic Differential Equations in Infinite Dimensional Space, Society for Industrial and Applied Mathematics, Philadelphia.
Jacod, J., and Shiryaev, A. N. (1987). Limit Theorem for Stochastic Processes, Springer-Verlag, Berlin.
Knapp, A. W. (1986). Representation Theory of Semisimple groups, Princeton University Press, New Jersey.
Kuo, H. H. (1975). Gaussian Measures in Banach Spaces, Lect. Notes in Math. Vol. 467, Springer, Berlin.
Malliavin, P. (1997). Stochastic Analysis, Springer-Verlag, Berlin.
Segal, I. E. (1962). Mathematical characterization of the physical vacuum, III. J. Math. 6, 500–523.
Segal, I. E. (1978). The complex wave representation of free Bose field, In Gohberg, I., and Kac, M., (eds.), Topics in Functional Analysis, Academic Press, New York, pp. 321–343.
Shigekawa, I. (1991). Itô Wiener expansion of holomorphic functions on the complex Wiener space, In Mayer, E., et al., (eds.), Stochastic Analysis, Academic Press, pp. 459–472.
Sugita, II. (1994). Properties of holomorphic Wiener functions—skeleton, contraction and local Taylor expansion. Prob. Th. Rel. Fields 100, 117–130.
Ungar, A. (1994). The abstract complex Lorentz transformation group with real metric, I. J. Math. Phys. 35, 1408–1426.
Yamazaki, Y. (1986). Measures on Infinite Dimensional Space, World Scientific, Singapore.
Rights and permissions
About this article
Cite this article
Luo, S. A Nonlinear Deformation of Wiener Space. Journal of Theoretical Probability 11, 331–350 (1998). https://doi.org/10.1023/A:1022623603891
Issue Date:
DOI: https://doi.org/10.1023/A:1022623603891