Invertibility of adapted perturbations of the identity on abstract Wiener space

Let (W,H,μ) be an abstract Wiener space. It is well known that a continuously increasing sequence of projections on H enables to define the notion of adapted shift. Under the assumption that such a sequence exists, we study the invertibility of adapted shifts on abstract Wiener space. In particular we extend a recent result of Üstünel which relates the invertibility of an adapted perturbation of the identity on the classical Wiener space, to the equality between the energy of the signal and the relative entropy of the measure it induces. We also extend this result to a probability absolutely continuous but not necessarily equivalent to the Wiener measure, with finite entropy. Finally, we relate this theorem both to the Monge problem, and to the innovation conjecture.