Minimal model of diffusion with time changing Hurst exponent

TL;DR

This paper introduces the incremental multifractional Brownian motion (IMFBM), a simple stochastic process modeling diffusion with time-varying Hurst exponent and diffusivity, useful for describing dynamic anomalous diffusion in changing environments.

Contribution

The paper presents the IMFBM process, deriving its mean squared displacement and correlations, and provides methods for estimation, advancing modeling of diffusion with evolving parameters.

Findings

Derived elementary formulas for MSD and correlations of IMFBM.

Validated IMFBM through simulations.

Developed estimation methods for IMFBM parameters.

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.