Given a stochastic process, I know how to find its covariance function, but is there anything available for the inverse?
For instance, https://en.wikipedia.org/wiki/Covariance_function#Parametric_families_of_covariance_functions lists 2 parametric functions:
$$C(x,y)=\sigma^2 e^{-d(x,y)}$$
and
$$C(x,y)=\sigma^2 e^{-d^2(x,y)}$$
for some distance function $d(x,y)$.
It says as well that the former generates non-smooth paths, and the latter smooth paths.
What processes have such covariance functions? and how can I estimate the implied smoothness of the paths?