How to show a stochastic process has infinitely differentiable sample paths

Question

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.

Answer 1

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.