  We study the thermal properties of a pinned disordered harmonic chain weakly
perturbed by a noise and an anharmonic potential. The noise is controlled by a
parameter $\lambda \rightarrow 0$, and the anharmonicity by a parameter
$\lambda' \le \lambda$. Let $\kappa$ be the conductivity of the chain, defined
through 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 for
which $\kappa \sim 1/\lambda$, and so shows the persitence of localization
effects for a non-integrable dynamics.
