Analytic Functions, Cauchy Formula, and Stationary Phase on a Real Abstract Wiener Space

A new complexification of a real abstract Wiener space will be introduced, and some analogs of the algebra of analytic functions on finite dimensional Euclidean space will be considered; analytic functions on the original space, their holomorphic prolongation to the complexified space, and holomorphic functions and a Cauchy formula on the complexified space. The Cauchy formula is a key tool to study probabilistic quantities via “deformation of the contour integration.” Namely, it will be applied to establish (i) an explicit representation of stochastic oscillatory integrals with quadratic phase function and (ii) a stationary phase estimation of the integrals. Further, the later estimation is applicable to study Gevrey type smoothness of density functions. An integration by parts formula on a totally real submanifold in the complexified space is also studied.