For a one dimensional Kolmogorov-Smirnov test there is a theoretical result, one can compare his or her test results directly by theoretical one, which is infinite sum (notice this formula is for big samples http://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test ), and this distribution is called a Kolmogorov distribution. And one can compute the significance level (p-value) directly from this CDF formula. Also it's possible to get a probability density distribution from this formular and many other things.
So my question : Is there a theoretical result for 2 dimensional distributions ? Or, if I compute a 2 dimensional kolmogorov-smirnov test myself, by code that I have written myself, how can I check the quality of my test result? In this case I would prefer to have theoretical information.