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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.

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I'm afraid I don't have a good answer to your question, but I believe the first black noises were not the Arratia flow (Brownian web). They appeared earlier in Tsirelson and Vershik's "Examples of nonlinear continuous tensor products of measure spaces and non-Fock factorizations" (although admittedly they were not noises on $\mathbb{R}$, if I recall correctly). Furthermore your definition of black is not quite correct. A noise is non-classical if it is not isomorphic to standard Gaussian white noise, and black if it has no classical subnoise. –  Tom Ellis Nov 18 '13 at 5:41
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