White noise: what is it, roughly?

In some sense, a white noise is the derivative of a Brownian motion (even though the Brownian motion is nowhere differentiable...)

Brownian motion: what is it, roughly?

Imagine a random walk that jumps every 0.01 s (second) by 0.1 m (meter) to the left or to the right, at random, with probabilities 0.5, 0.5. One second after starting (at the origin), it is located at a random point between -10 m and +10 m. Its mean (expected) value is 0; its standard deviation is 1 m; and its probability distribution is close to normal. Ignore the discreteness in space and time, and you get some idea of the simplest Brownian motion.

Scaling limit: what is it, roughly?

Consider a random walk as above; replace 0.1 m by some value h and 0.01 s by h²; let h tend to 0; then, the limit of the random walk is the (simplest) Brownian motion. That is the most famous example of scaling limit.

Black noise: what is it, roughly?

The random walk chooses (at each step) either "up" or "down". We may interprete "up" as a map (x to x+0.1 m) and "down" as another map (x to x-0.1 m). However, we may consider another pair of maps. The corresponding scaling limit (if exists) can sometimes (for some pairs of maps) be described via white noise. For some other pairs of maps, it cannot. Its derivative is still a noise in some sense, but not a classical noise. If it contains no white component, we call it black.

Black noise: is it a special case of a colored noise?

No, it is not. A colored noise is correlated; that is, its restrictions to disjoint intervals (of time) need not be independent. In contrast, both white and black noises are uncorrelated. In terms of Fourier transform: the spectral density of a white noise is constant; when it is non-constant, the noise is colored. However, a black noise has no spectral density at all! Being intrinsically nonlinear, it cannot be described via the (linear) Fourier transform.

In other words, the white noise excites linear sensors of all frequencies to the same extent; a colored noise excites them differently. A black noise does not excite them at all. Only nonlinear sensors are sensitive to black noises.

What is all that good for?

The very idea of scaling limit is important, since continuous models lead to simpler formulas and fewer parameters. (Just compare, say, a binomial distribution and the normal distribution...) In reality, Brownian motion is (nearly) piecewise linear on micro scale, which is intentionally ignored by macro theory. Likewise, a queue is discrete in reality, but continuous in fluid or diffusion approximation. The same happens to mathematical finance.

Percolation theory is quite difficult. Existence of its scaling limit is a long-standing open question. An impressive progress is made recently. Due to conformal invariance, the continuous model leads to amusing explicit formulas.

During half a century, the white noise was treated as the only possible scaling limit for a sequence of independent equiprobable random signs. Accordingly, white and Poisson noises (and their combinations) were treated as the only possible (uncorrelated) noises. Now we understand that classical results hold under a restrictive assumption of linearity. Classical independent increment processes take on values in a commutative group (R, R^d, or even a Hilbert space). Non-smooth stochastic flows, being noncommutative Brownian motions, can lead to nonclassical noises, which is quite interesting for turbulence theory.

Stability and sensitivity of Boolean (and other) functions, important for computer science, is closely related to noises. Namely, classical noises are described by stable functions; black noises - by sensitive functions.

A pair of independent random variables corresponds to the product of two probability spaces. Similarly, a noise (uncorrelated) corresponds to a continuous product of probability spaces, - still a mysterious object. Their L₂ spaces form a continuous tensor product of Hilbert spaces, called also a product system, - also a mysterious object in the theory of C^* algebras. Several open questions, motivated mostly by quantum field theory, were solved recently via nonclassical noises.