I am looking into limiting distributions (LD) of random walks on some 2D surfaces. In particular, I am interested in the LD of a random walk on a torus, Klein bottle, cylinder and Mobius strip, but I can't seem to find much on this online. I would assume that the LD of the torus is uniform as the torus is $S^1 \times S^1$ and the the LD of the random walk on the circle is uniform. Is this correct? I am not sure what to expect for the other surfaces.
I was wondering if anyone knows any good references on the LD's of random walks on these surfaces?
Many thanks!