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I would like to know if there is a natural notion of Gaussian white noise that has been defined on the real projective space $\mathbb{R}\mathbb{P}^n$ of dimension $n$.

My impression is that it is doable, for instance as a limit of homogeneous polynomial functions of increasing degree, but I could not find such a construction.

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  • $\begingroup$ What properties should this notion have, in your opinion? $\endgroup$
    – Igor Rivin
    Nov 6, 2016 at 20:57

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This is a well investigated issue in probability. To any Riemann metric on $\mathbb{RP}^n$ you can canonically associate a Gaussian white noise. Have a look at the beautiful book of Gelfand and Vilenkin "Generalized Functions, vol. 4: Applications of Harmonic Analysis".

The white noise is a generalized process with independent values at every point.

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