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In this work we study the necessary and sufficient conditions for a positiverandom variable whose expectation under the Wiener measure is one, to berepresented as the Radon-Nikodym derivative of the image of the Wiener measureunder an adapted perturbation of identity with the help of the associatedinnovation process. We prove that the innovation conjecture holds if and onlyif the original process is almost surely invertible. We also give variationalcharacterizations of the invertibility of the perturbations of identity and therepresentability of a positive random variable whose total mass is equal tounity. We prove in particular that an adapted perturbation of identity$U=I_W+u$ satisfying the Girsanov theorem, is invertible if and only if thekinetic energy of $u$ is equal to the entropy of the measure induced with theaction 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. Therelations with the Monge-Kantorovitch measure transportation are also studied.An application of these results to a variational problem related to largedeviations is also given.

Entropy, Invertibility and Variational Calculus of the Adapted Shifts on Wiener space