TITLE: Do there exist (almost surely) $C^{\infty}$-smooth Gaussian random fields?
QUESTION [5 upvotes]: 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?

REPLY [2 votes]: 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.