Invariance of Malliavin Fields on Itoʹs Wiener Space and on Abstract Wiener Space

Let (E, H, m) be an abstract Wiener space and (Ω, H, γ) be the corresponding Itoʹs Wiener space whereΩconsists of all the linear (but not necessarily continuous) functionals on the Hilbert spaceH. We show that one can always linearly embed (E, H, m) into (Ω, H, γ) in such a way that the family of allγ-regular measures onΩare exactly the family of the extensions of all probability measures of finite energy onE. A subsetAofEis a slim set if and only if it is a M-null set inΩ. The family of all MalliavinTr-fields onEare exactly the family of all the restrictions of MalliavinTr-fields onΩ. Moreover, the one to one mapping between Malliavin fields onΩand those onEis commutable with the gradient operator and keeps the Sobolev norms invariant. Hence most of the results of Malliavin calculus known for abstract Wiener space can be transferred to the Itoʹs Wiener space and vice versa.