Power Spectral Density of Brownian Motion despite non-stationary
Please see N.J. Kasdin, "Discrete Simulation of Colored Noise and Stochastic Processes and $1/f^\alpha$ Power Law Noise Generation", Proceedings of the IEEE, 83(5), (1995), 802-827. On page 806, Kasdin discusses the factor of 2 issue and states that "It is this author's belief that the factor of two in (24) is an error due to a broad-band bias introduced by the rectangular window." With a Hanning window (with 8/3 normalization), Kasdin gets an expression (equation 25) that converges to $1/\omega^2$ for large T. He also says "Since the formal spectral density of Brownian motion doesn't exist, the better question is which provides the most accurate information about the second order properties and possible linear models (see Section II-D) for Brownian motion?" Hope this helps a bit!