# Regularity of Gaussian process sample paths

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."?

## 1 Answer

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 $$C^{k_0}$$.