Tests for spatial stationarity (homogeneity)? There are many models for spatial point patterns and spatial marked point patterns that assume spatial homogeneity or stationarity. 
i) Is there a statistical test for determining this, where the null is that of stationarity?  
ii) Is there a procedural way of looking at the data to gauge/have a say at this, by say using variograms etc? 
 A: Three comments based on a mixture of experience and prejudice: 


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*What should be important here is that the researcher's substantive knowledge (or that of a collaborator), which may make the question obvious at some level. That is, it may be foolish to apply models assuming stationarity if there are known to be gross trends across a region that are important for the variable(s) being modelled. At a minimum, expect flak from experts if your application is a real stretch. 

*Nonstationarity may well be evident by fitting a model and then assessing the fit, e.g. if the fit is lousy, nonstationarity may be a likely suspect. But as often in statistics, an oversimplified model that is only a crude approximation may still be of use or interest. 

*Nonstationarity may be evident by inspection of basic maps, etc. 
In short, this answer stresses the scope for considering the answer informally as well as by seeking formal tests. "Informally" does include ensuring that subject-matter knowledge and expertise play a key part. 
A: Leung, Mei, and Zhang have developed two tests for whether GWR is a better fit than OLS regression. Their paper is here, but it's behind a paywall if you don't have academic access. As for variograms, etc. I know that Bivand, et al. cover tools and mechanisms In their book. I know a pdf of this exists because I have it, but I forgot where I got it from.
As far as spatial stationarity in general, I'm a little bit skeptical; GWR is the only method I have looked at specifically, but it seems to give contradictory answers perhaps because of its susceptibility to collinearity. I don't know what your particular application is, but in home price hedonics there has been some movement to autoregressive models that incorporate heterogeneity sympathetically (like the spatial Durbin model).
