I've been playing around with some random field models and noticed that the apparent differentiability seems to be related to the covariance function's behavior at 0. My initial guess was that if $\frac d {dx} C(x) = 0$, then a resulting random field would be differentiable almost everywhere. However, I got stuck trying to prove this.
So, is this a true statement? Is it an understatement? That is, would that condition imply smoothness a.e.? If the statement is true, then is the condition necessary as well as sufficient?
I realize I may be off entirely here. I suppose the answer could depend on the distribution (Gaussian/something else) or the topological space. Are there any well known conditions that determine the smoothness of a random field?
*This isn't a homework problem. I'm just curious about this.
