The realization of positive random variables via absolutely continuous transformations of measure on Wiener space

Let $\mu$ be a Gaussian measure on some measurable space $\{W = \{w\}, \mathcal{B}(W)\}$ and let $\nu$ be a measure on the same space which is absolutely continuous with respect to $\nu$. The paper surveys results on the problem of constructing a transformation $T$ on the $W$ space such that $Tw = w + u(w)$ where $u$ takes values in the Cameron-Martin space and the image of $\mu$ under $T$ is $\mu$. In addition we ask for the existence of transformations $T$ belonging to some particular classes.