solution of the equation with the help of the Duhamel formula, see . The corresponding representation for the Wigner distribution is given in . Having already established the macroscopic limit for the dynamics with no stochastic noise, we can use the Duhamel representation to identify the kinetic limit of the noisy microscopic dynamics when the thermostat temperature $T=0$, see Theorem \[cor020805-19\]. The extension to the case when the temperature $T>0$ is possible by another application of the Duhamel formula, see Section \[sec10\].

Concerning the organization of the paper, Section \[prelim\] is devoted to preliminaries and the formulation of the main result, see Theorem \[main-thm\]. Among things discussed is the rigorous definition of a solution of a kinetic equation with the interface condition , see Sections \[sec2.6.3\] and \[sec2.6.4\]. Section \[sec3\] deals with the basic properties of the microscopic dynamics obtained by removal of stochastic noises, both between the particles of the chain and the thermostat. This dynamics is an auxiliary tool for the mild formulation of the microscopic dynamics corresponding to the chain with inter-particle scattering and thermostat. We discuss first the case when thermostat is set at $T=0$, see Section \[sec4\]. In this section we obtain also basic estimates 