Since the seminal work of Wiener, the chaos expansion has evolved to a
powerful methodology for studying a broad range of stochastic differential
equations. Yet its complexity for systems subject to the white noise remains
significant. The issue appears due to the fact that the random increments
generated by the Brownian motion, result in a growing set of random variables
with respect to which the process could be measured. In order to cope with this
high dimensionality, we present a novel transformation of stochastic processes
driven by the white noise. In particular, we show that under suitable
assumptions, the diffusion arising from white noise can be cast into a
logarithmic gradient induced by the measure of the process. Through this
transformation, the resulting equation describes a stochastic process whose
randomness depends only upon the initial condition. Therefore the stochasticity
of the transformed system lives in the initial condition and thereby it can be
treated conveniently with the chaos expansion tools
