Fractional Brownian Motion & Anomalous Diffusion

Fractional Brownian motion (fBm) with Hurst exponent H controls memory and diffusion scaling. H<0.5: antipersistent subdiffusion (negatively correlated increments). H=0.5: standard Brownian motion. H>0.5: persistent superdiffusion (positively correlated).

Hurst H 0.50

N steps 500

N trajectories 6

H =

0.50

MSD exponent

2H = 1.00

Diffusion type

Normal

Increment corr

MSD: ⟨|X(t) - X(0)|²⟩ ~ t^(2H) | Increment autocorr: C(k) = ½[(k+1)^(2H) - 2k^(2H) + |k-1|^(2H)]
Fractional calculus: fBm = B_H(t) = ∫ K_H(t,s) dB(s) where K_H is Molchan-Golosov kernel.