Since the seminal work of Wiener (Am J Math 60:897–936, 1938), 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 on the initial condition. Therefore, the stochasticity of the transformed system lives in the initial condition and it can be treated conveniently with chaos expansion tools.

自维纳（Wiener）的开创性工作（Am J Math 60：897–936，1938）以来，混沌扩展已发展成为一种用于研究各种随机微分方程的强大方法。然而，对于遭受白噪声的系统而言，其复杂性仍然很大。出现该问题的原因是，布朗运动产生的随机增量导致可以测量过程的一组随机变量不断增加。为了应对这种高维度，我们提出了一种由白噪声驱动的随机过程的新型变换。尤其是，我们表明，在适当的假设下，由白噪声引起的扩散可以转换为由该过程的度量引起的对数梯度。通过这种转变，结果方程描述了一个随机过程，其随机性仅取决于初始条件。因此，转换后系统的随机性处于初始状态，并且可以使用混沌扩展工具方便地进行处理。