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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.