  Motion in bounded domains represents a paradigm in several settings: from
billiard dynamics, to random walks in a finite lattice, with applications to
relevant physical, ecological and biological problems. A remarkable universal
property, involving the average of return times to the boundary, has been
theoretically proposed, and experimentally verified in quite different
contexts. We discuss here mechanisms that lead to violations of universality,
induced by boundary effects. We suggest that our analysis should be relevant
where non homogeneity appears in the stationary probability distribution in
bounded domain.
