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Let $M$ be an arbitrary smooth manifold of dimension $n$. For simplicity, let's assume that $M$ is boundary-less. Can we construct a gaussian random field on $M$?

If the result is not true for arbitrary $M$, is there a theorem that provides minimal assumptions on $M$ for which we always can find a gaussian random field?

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  • $\begingroup$ The existence is certainly true for any manifold: just take a constant field. $\endgroup$ – zhoraster Oct 21 '15 at 6:15
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Yes. Robert Adler has written some books on Gaussian random fields that I enjoyed perusing. For specific examples of Gaussian random fields that are guaranteed to exist on closed manifolds under minimal assumptions you could look at http://arxiv.org/abs/1207.6419 and the references therein.

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