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

Abstract

Let $μ$ be a Gaussian measure on some measurable space $\{W=\{w\},{\mathcal{B}}(W)\}$ and let $ν$ be a measure on the same space which is absolutely continuous with respect to $ν$. 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 $μ$ under $T$ is $μ$. In addition we ask for the existence of transformations $T$ belonging to some particular classes.