We propose a generalization of the widely used fractional Brownian motion (FBM), memory-multi-FBM (MMFBM), to describe viscoelastic or persistent anomalous diffusion with time-dependent memory exponent α(t) in a changing environment. In MMFBM the built-in, long-range memory is continuously modulated by α(t). We derive the essential statistical properties of MMFBM such as its response function, mean-squared displacement (MSD), autocovariance function, and Gaussian distribution. In contrast to existing forms of FBM with time-varying memory exponents but a reset memory structure, the instantaneous dynamic of MMFBM is influenced by the process history, e.g., we show that after a steplike change of α(t) the scaling exponent of the MSD after the α step may be determined by the value of α(t) before the change. MMFBM is a versatile and useful process for correlated physical systems with nonequilibrium initial conditions in a changing environment.
W. Wang et al., "Memory-multi-fractional Brownian Motion With Continuous Correlations," Physical Review Research, vol. 5, no. 3, article no. L032025, American Physical Society, Jul 2023.
The definitive version is available at https://doi.org/10.1103/PhysRevResearch.5.L032025
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01 Jul 2023