Snoek et al, have a recent paper "Input Warping for Bayesian Optimization of Non-Stationary Functions" (http://arxiv.org/abs/1402.0929) which mentions "stationary functions".
I understand what a stationary kernel or process is, but I can't find a definition of a stationary function. I kind of suspect there is no such thing, and they are using the term informally to mean something along the lines of "a function such that any epsilon-ball of the function has non-zero probability according to some stationary process". But the top example in figure 1 (sine wave) doesn't seem to fit this description to me, since the distance between the image of two points themselves a fixed distance apart changes dramatically as you move the points.
I guess you can maybe define a process whose value is fixed at the peaks and troughs of the sine wave, and then this might be a stationary function under that process?
Or am I totally wrong about the definition?