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 response function, mean-squared displacement (MSD),
autocovariance function, and Gaussian distribution. In contrast to existing
forms of FBM with time-varying memory exponents but reset memory structure, the
instantaneous dynamic of MMFBM is influenced by the process history, e.g., we
show that after a step-like 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 non-equilibrium initial conditions in a changing
environment.Comment: 15 pages, 10 figures, RevTe