  We consider a finite region of a $d$-dimensional lattice, $d\in\mathbb{N}$,
of weakly coupled harmonic oscillators. The coupling is provided by a
nearest-neighbour potential (harmonic or not) of size $\varepsilon$. Each
oscillator weakly interacts by force of order $\varepsilon$ with its own
stochastic Langevin thermostat of arbitrary positive temperature. We
investigate limiting as $\varepsilon\rightarrow 0$ behaviour of solutions of
the system and of the local energy of oscillators on long-time intervals of
order $\varepsilon^{-1}$ and in a stationary regime. We show that it is
governed by an effective equation which is a dissipative SDE with nondegenerate
diffusion. Next we assume that the interaction potential is of size
$\varepsilon\lambda$, where $\lambda$ is another small parameter, independent
from $\varepsilon$. Solutions corresponding to this scaling describe small low
temperature oscillations. We prove that in a stationary regime, under the limit
$\varepsilon\rightarrow 0$, the main order in $\lambda$ of the averaged
Hamiltonian energy flow is proportional to the gradient of temperature. We show
that the coefficient of proportionality, which we call the conductivity, admits
a representation through stationary space-time correlations of the energy flow.
Most of the results and convergences we obtain are uniform with respect to the
number of oscillators in the system.
