Existence and uniqueness of an invariant measure for a chain of oscillators in contact with two heat baths

In this note we consider a chain of $N$ oscillators, whose ends are in contact with two heat baths at different temperatures. Our main result is the exponential convergence to the unique invariant probability measure (the stationary state). We use the Lyapunov's function technique of Rey-Bellet and coauthors with different model of heat baths, and adapt these techniques to two new case recently considered in the literature by respectively Bernardin and Olla, Lefevere and Schenkel