Langevin Dynamics with a Tilted Periodic Potential

We study a Langevin equation for a particle moving in a periodic potential in the presence of viscosity γ and subject to a further external field α. For a suitable choice of the parameters α and γ the related deterministic dynamics yields heteroclinic orbits. In such a regime, in absence of stochastic noise both confined and unbounded orbits coexist. We prove that, with the inclusion of an arbitrarly small noise only the confined orbits survive in a sub-exponential time scale.