Thanks for your help in advance. I'm interested in understanding the properties of derivatives of a differentiable stationary Gaussian process. Specifically, is the derivative also a Gaussian process?
The answer is yes in the sense that the gradient of the mean is a GP defined jointly with the original GP. I'm sure it's discussed elsewhere, but you can find derivations in section 5 of http://www.biostat.umn.edu/~sudiptob/ResearchPapers/BGjasa06.pdf.
Malliavin calculus is the apropriate framework for your question Take $F \in \mathscr S$ (the space of functions such that all derivatives are of polynomial growth).
We define $$ DF = \sum_i \partial_i f ( W(h_1), \dots, W(h_n) ) h_i, $$ and this should be regarded as an $H=L^2$-valued r.v.
$D$ is well-defined. In particular for $F=W(h)=\int_0^1 h_t d B_t$ this is a consequence of the Ito-isometry.