We study the model of a strongly non-linear chain of particles coupled to two heat baths at different temperatures. Our main result is the existence and uniqueness of a stationary state at all temperatures. This result extends those of Eckmann, Pillet, Rey-Bellet [EPR99a, EPR99b] to potentials with essentially arbitrary growth at infinity. This extension is possible by introducing a stronger version of Hormander's theorem for Kolmogorov equations to vector fields with polynomially bounded coefficients on unbounded domains. Introduction In this paper, we study the statistical mechanics of a highly non-linear chain of oscillators coupled to two heat reservoirs which are at (arbitrary) different temperatures. We show that such systems have, under suitable conditions, a unique stationary state, in which heat flows from the hotter reservoir to the cooler one. These results are an extension of the same statements obtained by Eckmann, Pillet and ReyBellet in [EPR99a, EPR99b] where it was ass...