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We study the thermal properties of a pinned disordered harmonic chain weaklyperturbed by a noise and an anharmonic potential. The noise is controlled by aparameter $\lambda \rightarrow 0$, and the anharmonicity by a parameter$\lambda' \le \lambda$. Let $\kappa$ be the conductivity of the chain, definedthrough the Green-Kubo formula. Under suitable hypotheses, we show that $\kappa= \mathcal O (\lambda)$ and, in the absence of anharmonic potential, that$\kappa \sim \lambda$. This is in sharp contrast with the ordered chain forwhich $\kappa \sim 1/\lambda$, and so shows the persitence of localizationeffects for a non-integrable dynamics.

Small perturbation of a disordered harmonic chain by a noise and an anharmonic potential