It is possible to model dissipation effects subjected by a particle by interactions between the particle and its environment. This seminal idea dates back to Caldeira-Leggett in the ’80ies. The specific case of a particle interacting with vibrational degrees of freedom has been thoroughslyi investigated by S. De Bi`evre and his collaborators. We will go back to theseissues in the framework of kinetic equations, and also consider quantum ver-sions of the problem based on couplings with the Schr ̈odinger equation. We are particularly interested in stability issues. We will describe; through rigorous statements and numerical experiments, analogies and differences with the case of a single classical particle and with the standard coupling with thePoisson equation.