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Let $d \ge 1$. Do there exist Gaussian random fields on $\mathbb R^d$ which are (almost surely) $C^{\infty}$-smooth, but which are not analytic? If so, what are necessary and sufficient conditions on the covariance function to assure that the field is (a.s.) $C^{\infty}$-smooth? |
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I think, you get what you want if you convolve nonsmooth trajectories (like those of a Wiener process) with a $C^\infty$ kernel that is not analytic. |
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