Detecting subtle deviations in Brownian motion representations driven by a Schauder basis

In this study, we construct surrogate stochastic processes that are challenging to distinguish from ordinary Brownian motion using a method based on the Schauder representation. Specifically, by assuming non-Gaussian (beta and uniform) distributions for the Schauder coefficients, we generate sample paths that preserve key properties of Brownian motion-such as quadratic variation, covariance structure, pointwise Hölder regularity, uncorrelated increments, as well as Gaussian marginal distributions. However, a deeper analysis relying on entropy-based measures and sliding-window spectral variance reveals that only the Gaussian-based construction preserves the expected randomness and the consistent spectral behavior of Brownian motion over time. In contrast, non-Gaussian variants exhibit subtle deviations from true Brownian motion.