Minimal model of diffusion with time changing Hurst exponent

Abstract

We introduce the stochastic process of incremental multifractional Brownian motion (IMFBM), which locally behaves like fractional Brownian motion with a given local Hurst exponent and diffusivity. When these parameters change as function of time the process responds to the evolution gradually: only new increments are governed by the new parameters, while still retaining a power-law dependence on the past of the process. We obtain the mean squared displacement and correlations of IMFBM which are given by elementary formulas. We also provide a comparison with simulations and introduce estimation methods for IMFBM. This mathematically simple process is useful in the description of anomalous diffusion dynamics in changing environments, e.g., in viscoelastic systems, or when an actively moving particle changes its degree of persistence or its mobility.