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. 

