Mixing Times and Moving Targets

We consider irreducible Markov chains on a nite state space. We show that the mixing time of any such chain is equivalent to the maximum, over initial statesx and moving large sets (As)s, of the hitting time of (As)s starting from x. We prove that in the case of the d-dimensional torus the maximum hitting time of moving targets is equal to the maximum hitting time of stationary targets. Nevertheless, we construct a transitive graph where these two quantities are not equal, resolving an open question of Aldous and Fill on a \cat and mouse" game.