Energy growth for systems of coupled oscillators with partial damping

Abstract We consider two interacting particles on the circle. The particles are subject to stochastic forcing, which is modeled by white noise. In addition, one of the particles is subject to friction, which models energy dissipation due to the interaction with the environment. We show that, in the diffusive limit, the absolute value of the velocity of the other particle converges to the reflected Brownian motion. In other words, the interaction between the particles is asymptotically negligible in the scaling limit. The proof combines averaging for large energies with large deviation estimates for small energies.