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Consider a Gaussian process on $[0,1]$ given by a kernel function $K: [0,1]^2\to\mathbb{R}$. Under what conditions can we conclude that the sample paths are $C^k$ with probability 1?

This question is partially motivated by section 2.2 of these (http://galton.uchicago.edu/~lalley/Courses/385/Old/GaussianProcesses.pdf) notes, in which it is remarked that "In general, the degree of smoothness of a Gaussian process is determined by the smoothness of its covariance function near the diagonal", although a general theorem along these lines is not proven in the notes.

So is there a known result along the lines of "If the covariance function is $C^k$ along the diagonal, then the sample paths are $C^k$ a.s."?

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This paper provides general results on the smoothness of sample paths of second order random fields. In particular, if the covariance is $C^k$ near the diagonal, the sample paths a.s lie in the local Sobolev space of order $k$. So, by the Sobolev embedding theorem, given $k_0$, there exist $k_1 > k_0$ such that if the covariance is $C^{k_1}$ near the diagonal, the sample paths are a.s. in $C^{k_0}$.

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