the limit is to introduce the following interface conditions at $y=0$ on : $$\label{bc-int}
  \begin{split}
W(t,0^+,k) &=p_-(k) W(t,0^+, -k) + p_+(k)W(t, 0^-,k)+{{\fgeeszett}}(k)T, \quad\hbox{ for $0< k\le 1/2$},\\
W(t,0^-, k) &=p_-(k)W(t,0^-,-k) + p_+(k) W(t,0^+, k) +{{\fgeeszett}}(k)T,\quad \hbox{ for $-1/2< k< 0$.}
\end{split}$$ Interpreting $W(t,y,k)$ as the density of the energy of the phonons of mode $k$ at time $t$ and position $y$, then $ p_+(k)$, $ p_-(k)$ and ${{\fgeeszett}}(k)$ are respectively the probabilities for transmission, reflection and absorption of a phonon of mode $k$ when it crosses $y=0$, while ${{\fgeeszett}}(k)T$ is the rate of creation of a phonon of that mode. These probabilities are functions of $k$ and depend only on the dispersion relation $\omega(\cdot)$ and the intensity $\gamma_0$ of the thermostat (cf. ). They are properly normalized, i.e. $ p_+(k) + p_-(k) + {{\fgeeszett}}(k) = 1$, so that $W(t,y,k) = T$ is a stationary solution (thermal equilibrium).

This result was recently proven in the absence of the conservative noise in the bulk (i.e. $\gamma_0= 0$ in ), see [@kors]. Then, the resulting dynamics outside the interface, given by , reduces itself to pure transport as ${\gamma}_0=0$. Obviously, the coefficients appearing in the interface conditions do not depend on the presence 