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

Abstract

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.

Published at http://dx.doi.org/10.1214/154957806000000069 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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