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?
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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 welldefined. In particular for $F=W(h)=\int_0^1 h_t d B_t$ this is a consequence of the Itoisometry. 

