Fractional Ornstein–Uhlenbeck Noise

📐 Definition

Fractional Ornstein–Uhlenbeck (fOU) processes extend the classic OU process by driving it with fractional Brownian motion, blending mean reversion with persistent or antipersistent correlations.

In the mean-zero case, a common (mild) form is

X_t = X_0 e^{-\lambda t} + \sigma\int_0^t e^{-\lambda(t-s)}\,dB_H(s)

where the stochastic integral is interpreted in a manner appropriate to $H$ (e.g. Itô for $H=\tfrac12$ , and pathwise/Young-type for $H>\tfrac12$ ).

Domain and Codomain

Defined over continuous or discretized time; outputs are real-valued stochastic trajectories.

⚙️ Key Properties

\tau_{\text{relax}} = \lambda^{-1}

Mean reversion rate $\lambda$ sets decay to equilibrium; the Hurst parameter $H$ shapes memory, with $H>1/2$ yielding persistence and $H<1/2$ yielding anti-persistence (and smaller $H$ producing rougher sample paths).

$H = \tfrac{1}{2}$ recovers the classical Ornstein–Uhlenbeck process (in which case $B_{1/2}$ is standard Brownian motion and the SDE is interpreted in the Itô sense). Formally taking $\lambda \to 0$ yields $X_t \to X_0 + \sigma B_H(t)$ ; large $\lambda$ forces rapid return toward the mean-reverting equilibrium.

🎯 Special Cases and Limits

$H=\tfrac12$ recovers the classical OU process.
$\lambda\to 0$ approaches fractional Brownian motion driving.
Large $\lambda$ yields fast relaxation with reduced variance.

The standard Ornstein–Uhlenbeck process is the $H=\tfrac{1}{2}$ case; fractional Brownian motion is the $\lambda=0$ driver; Gaussian white noise arises as the derivative of Brownian motion and underpins the $H=\tfrac{1}{2}$ forcing.

Usage in Oakfield

Oakfield uses “fractional OU” behavior in its colored-noise operator:

stochastic_noise operator generalizes OU updates with a spectral exponent parameter (alpha) to produce colored noise (OU-like when alpha≈1, with pink/blue tilts as alpha varies).
This operator is used as a reusable forcing source in operator graphs rather than being tied to a single integrator.

Historical Foundations

📜 Deforming OU with Fractional Drivers

Replacing Brownian motion with fractional Brownian motion is a standard way to introduce long-memory correlations into otherwise Markovian relaxation dynamics.

🌍 Modern Perspective

Fractional OU models are used when exponential correlation is too short-ranged and algebraic memory is required.

📚 References

Beran, Statistics for Long-Memory Processes
Samorodnitsky & Taqqu, Stable Non-Gaussian Random Processes (context on long-memory modeling)