  In this work we study the necessary and sufficient conditions for a positive
random variable whose expectation under the Wiener measure is one, to be
represented as the Radon-Nikodym derivative of the image of the Wiener measure
under an adapted perturbation of identity with the help of the associated
innovation process. We prove that the innovation conjecture holds if and only
if the original process is almost surely invertible. We also give variational
characterizations of the invertibility of the perturbations of identity and the
representability of a positive random variable whose total mass is equal to
unity. We prove in particular that an adapted perturbation of identity
$U=I_W+u$ satisfying the Girsanov theorem, is invertible if and only if the
kinetic energy of $u$ is equal to the entropy of the measure induced with the
action of $U$ on the Wiener measure $\mu$, in other words $U$ is invertible iff
$$ \half \int_W|u|_H^2d\mu=\int_W \frac{dU\mu}{d\mu}\log\frac{dU\mu}{d\mu}d\mu
>. $$ otherwise the l.h.s. is always strictly greater than the r.h.s. The
relations with the Monge-Kantorovitch measure transportation are also studied.
An application of these results to a variational problem related to large
deviations is also given.
