Transformations of a class of Gaussian processes to the Brownian motion are obtained by reproducing kernel Hilbert space methods. These transformations are such that the value of the transformed process at any point of time is given in terms of the sample path of the original process up to that time. In certain situations the boundary-crossing behaviors of the original process and the transformed process are related.
P. K. Bhattacharya. "Transformations of Gaussian Processes to the Brownian Motion." Ann. Math. Statist. 42 (6) 2008 - 2017, December, 1971. https://doi.org/10.1214/aoms/1177693068