Anomalous Diffusion & Hurst Exponent

Fractional Brownian motion (fBm) with Hurst exponent H generates anomalous diffusion: MSD ∝ t^(2H). When H=0.5, normal Brownian motion. H<0.5 gives subdiffusion (anti-persistent, mean-reverting). H>0.5 gives superdiffusion (persistent, trending). The spectrum of H captures phenomena from protein dynamics (H≈0.4) to market trends (H≈0.6) to turbulence (H≈0.33).

Hurst H 0.70 Trajectories 6 Steps N 256

Hurst exponent H = 0.70
MSD exponent: α = 1.40
Diffusion type: Superdiffusion

Regime classification:
H = 0.5 → Normal (Brownian)
H < 0.5 → Sub-diffusion
H > 0.5 → Super-diffusion

ACF of increments:
C(τ) = ½[(τ+1)^(2H) − 2τ^(2H) + |τ−1|^(2H)]
Positive (H>0.5) → persistence
Negative (H<0.5) → anti-persistence