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Let $(W,H,\mu)$ be the classical Wiener space, assume that $U=I_W+u$ is anadapted perturbation of identity satisfying the Girsanov identity. Then, $U$ isinvertible if and only if the kinetic energy of $u$ is equal to the relativeentropy of the measure induced with the action of $U$ on the Wiener measure$\mu$, in other words $U$ is invertible if and only if $$ \half\int_W|u|_H^2d\mu=\int_W \frac{dU\mu}{d\mu}\log\frac{dU\mu}{d\mu}d\mu . $$

A necessary and sufficient condition for the invertibility of adapted perturbations of identity on the Wiener space