Asymptotic spatial distribution in a simple 2D random walk

Hi:
I have a basic discrete-time random walk in a bounded 2D domain R: a point moves a fixed distance d at an angle A uniformly distributed over (0,2pi). Next step: same thing. (We'll ignore the tricky question of what to do at the border: some bouncing off rule will do).
Is there a result on the asymtptotic spatial distribution of the point, which I assume would be something along the lines of a uniform distribution over R?? Pointers and references from random walk experts would be most appreciated.
Marc Artzrouni
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