Let us suppose we have a centered Gaussian random field defined on the reals by the covariance structure. Is there a way of showing that this field has infinitely differentiable sample paths? Any theorem or result in this direction would be helpful.



How to show a stochastic process has infinitely differentiable sample paths
In general, you need the covariance to be smooth, especially near the diagonal. For some particular covariances there are very quick ways to prove what you want: for example if you can rewrite the field as a convolution of a random distribution with a mollifier. As for a general theorem, see Thm 2.2.3 in "Random Fields and Their Geometry" by R. J. adler and J. E. Taylor. This involves suitable bounds on a finite difference operator hitting both entries of the covariance. 


