Anticipating Flows on the Wiener Space Generated by Vector Fields of Low Regularity

Sobolev spaces for Banach space valued Wiener functionals are constructed using the operator norm for theH-derivatives. The corresponding Sobolev norms are shown to be strictly weaker than the usual Malliavin Sobolev norms. Existing results on flows on an abstract Wiener space generated by a vector field are generalized to the case where the vector field is a sum of a not necessarily bounded skew-adjoint operator and a non-linear part whose regularity is measured by the new Sobolev norms. This is equivalent to a non-linear perturbation of a semi-group of rotations.