A Nonlinear Deformation of Wiener Space

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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 L2-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.

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