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."?