Spectral design of anomalous diffusion

This paper poses and addresses the following goal: design a random motion that displays one type of anomalous diffusion in real space, and another type of anomalous diffusion in Fourier space. The two prominent Gaussian and selfsimilar anomalous-diffusion models – scaled Brownian motion (SBM), and fractional Brownian motion (FBM) – are not up to the task. This paper establishes a class of random motions that are up to the task. As a ‘bonus’, the random-motions’ velocities are also up to an additional task: generating flicker noise (1/f noise) behaviors. In general, the random motions are neither Gaussian nor selfsimilar. When both Gaussian and selfsimilar then the resulting random motion is a ‘hybrid’ Brownian model: it generalizes SBM on the one hand, and it generalizes the Riemann–Liouville FBM on the other hand.