With noise in the sense of i.i.d. random sequence, a noise is black if it is not isomorphic to standard Gaussian white noise.
Tsirelson showed the existence of black noise through the scaling limit of coalescing Brownian motion.
Watanabe gave a simpler example, also based on a coalescing stochastic flow, but using a singular diffusion.
Nelson, chap 18, showed how to construct a Brownian motion using non-standard analysis.
I'm wondering if someone tried to construct a black noise using non-standard method.