Memory-multi-fractional Brownian motion with continuous correlations

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 $\alpha \left(t\right)$ in a changing environment. In MMFBM the built-in, long-range memory is continuously modulated by $\alpha \left(t\right)$. 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 $\alpha \left(t\right)$ the scaling exponent of the MSD after the $\alpha$ step may be determined by the value of $\alpha \left(t\right)$ before the change. MMFBM is a versatile and useful process for correlated physical systems with nonequilibrium initial conditions in a changing environment.

Figure 1

Sample trajectories for MMFBM (1) (red) and MFBM (S6) (blue) for steplike protocol (3) for $\alpha \left(t\right)$ with switching time $\tau =50$ and (a) ${\alpha }_1=0.2$, ${\alpha }_2=1.8$; (b) ${\alpha }_1=1.8$, ${\alpha }_2=0.2$. In each panel both trajectories are based on the same realization of the parental Wiener process. For smooth protocol $\alpha \left(t\right)$, see Fig. S1. On a log-log scale the behavior for smooth and steplike protocol generally appear quite similar. Note the disparate behavior for $t>\tau$ in panel (b); see discussion below.

Figure 2

MSD for MMFBM $X\left(t\right)$ (1) for steplike and smooth protocol $\alpha \left(t\right)$ for two combinations of ${\alpha }_1$ and ${\alpha }_2$ (see legend). Full lines represent Eq. (5); symbols represent stochastic simulations. A comparison with MFBM is shown in Fig. S2 [76].

Figure 3

Numerical evaluations (lines) and simulations (symbols) for the ACVF $C(t,\mathrm{\Delta })$ at different times for MMFBM (1) with switching time $\tau =1$ and (a) ${\alpha }_1=1.8$, ${\alpha }_2=1.2$; (b) ${\alpha }_1=1.8$, ${\alpha }_2=0.2$. The inset in panel (b) shows the numerically obtained form close to $\mathrm{\Delta }=0$, demonstrating the antipersistence at long times $t$, when ${\alpha }_2=0.2$ becomes the dominant contribution. Note that in the main panel (b) the ACVF is shown in non-normalized form for better visibility.