Occupation times on a comb with ramified teeth

We investigate occupation time statistics for random walks on a comb with ramified teeth. This is achieved through the relation between the occupation time and the first passage times. Statistics of occupation times in half space follows Lamperti’s distribution, i.e. the generalized arcsine law holds. Transitions between different behaviors are observed, which are controlled by the size of the backbone and teeth of the comb, as well as bias. Occupation time on a non-simply connected domain is analyzed with a mean-field theory and numerical simulations. In that case, the generalized arcsine law isn’t valid.