Gaussian White Noise

📐 Definition

Gaussian white noise $\eta$ is a zero-mean generalized process with covariance

\mathbb{E}[\eta(t)\eta(s)] = \sigma^2 \delta(t-s)

interpreted in the distributional sense.

Domain and Codomain

$t \in \mathbb{R}$ (or $t \ge 0$ for causal models). Real-valued generalized process; finite-variance averages only after integration or discretization.

⚙️ Key Properties

Stationary with flat power spectral density $S(\omega) = \sigma^2$ for all $\omega \in \mathbb{R}$ . Independent increments after time integration: $\int_{t_0}^{t_1}\eta(t)\,dt \sim \mathcal{N}(0, \sigma^2(t_1-t_0))$ .

Discrete sampling with timestep $\Delta t$ yields $\eta_n \sim \mathcal{N}(0, \sigma^2/\Delta t)$ so that integrated variance matches the continuous model.

🎯 Special Cases and Limits

Time integration yields Brownian increments with variance proportional to the interval length.
Discretization requires scaling $\sigma^2/\Delta t$ to match continuous-time variance growth.

Ornstein–Uhlenbeck processes are filtered versions of $\eta$ ; Fourier transforms of white noise remain white in frequency.

Usage in Oakfield

Oakfield uses Gaussian white noise in two main places:

Stimulus operators: stimulus_white_noise writes seeded spatial white noise into real or complex fields (complex writes independent noise to Re/Im).
Stochastic integration: integrators can apply stochastic increments after each step (default noise is Gaussian; configurable to uniform or Laplace), and stochastic_noise provides an OU/fractional-OU style colored-noise operator.

Historical Foundations

📜 Brownian Motion and Generalized Derivatives

White noise is often formalized as the (generalized) time derivative of Brownian motion, making it a fundamental building block for stochastic differential equations.

🌍 Modern Perspective

It is the canonical idealization of uncorrelated forcing in time, with flat power spectral density.

📚 References

Øksendal, Stochastic Differential Equations
Gardiner, Stochastic Methods