Approach to the steady state in kinetic models with thermal reservoirs at different temperatures

We continue the investigation of kinetic models of a system in contact via stochastic interactions with several spatially homogeneous thermal reservoirs at different temperatures. Considering models different from those investigated in earlier work, we explicitly compute the unique spatially uniform non-equilibrium steady state (NESS) and prove that it is approached exponentially fast from any uniform initial state. This leaves open the question of whether there exist NESS that are not spatially uniform. Making a further simplification of our models, we then prove non-existence of such NESS and exponential approach to the unique spatially uniform NESS (with a computably boundable rate). The method of proof relies on refined Doeblin estimates and other probabilisitic techniques, and is quite different form the analysis in earlier work that was based on contraction mapping methods.