  We consider controlled random walks that are martingales with uniformly
bounded increments and nontrivial jump probabilities and show that such walks
can be constructed so that P(S_n^u=0) decays at polynomial rate n^{-\alpha}
where \alpha>0 can be arbitrarily small. We also show, by means of a general
delocalization lemma for martingales, which is of independent interest, that
slower than polynomial decay is not possible.
