Fractional Brownian Motion

📐 Definition

For $H \in (0,1)$, fractional Brownian motion $B_H$ is a zero-mean Gaussian process with covariance

$$\mathbb{E}\!\left[B_H(t) B_H(s)\right] = \tfrac{1}{2}\left(|t|^{2H} + |s|^{2H} - |t-s|^{2H}\right)$$

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Domain and Codomain

$t \ge 0$ (extendable to $\mathbb{R}$ by stationarity of increments). Real-valued with continuous sample paths almost surely.

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⚙️ Key Properties

Self-similarity: $B_H(ct) \stackrel{d}{=} c^{H} B_H(t)$ for $c>0$. Increments are stationary but correlated; correlation decays algebraically for $H \ne \tfrac{1}{2}$.

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$H = \tfrac{1}{2}$ reduces to standard Brownian motion. $H < \tfrac{1}{2}$ yields anti-persistent increments; $H > \tfrac{1}{2}$ yields persistent (long-memory) increments.

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🎯 Special Cases and Limits

• $H=\tfrac12$ is standard Brownian motion.
• $H<\tfrac12$ gives anti-persistent increments; $H>\tfrac12$ gives long-memory persistence.

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🔗 Related Functions

Fractional derivatives of $B_H$ produce fractional Gaussian noise; Ornstein–Uhlenbeck processes provide exponentially correlated alternatives.

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Usage in Oakfield

Oakfield provides fBm-style structure as a stimulus generator:

• stimulus_fbm operator generates fractal Brownian motion patterns parameterized by Hurst exponent and octave/lacunarity settings, with a seed for reproducibility.
• Used to seed multiscale structure in fields (as a stimulus), often paired with filters (sieve) or measurement operators.

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Historical Foundations

📜 Long-Range Dependence

Fractional Brownian motion provides a Gaussian model with tunable self-similarity and long-range dependence via the Hurst parameter.

🌍 Modern Perspective

It is widely used to model rough forcing and anomalous diffusion, where correlations decay algebraically rather than exponentially.

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📚 References

• Mandelbrot & Van Ness, “Fractional Brownian Motions, Fractional Noises and Applications” (1968)
• Beran, Statistics for Long-Memory Processes