Sub-exponential mixing of random billiards driven by thermostats

We study the class of open continuous-time mechanical particle systems introduced in the paper by Khanin and Yarmola (2013 Commun. Math. Phys. 320 121-47). Using the discrete-time results from Khanin and Yarmola (2013 Commun. Math. Phys. 320 121-47) we demonstrate rigorously that, in continuous time, a unique steady state exists and is sub-exponentially mixing. Moreover, all initial distributions converge to the steady state and, for a large class of initial distributions, convergence to the steady state is sub-exponential. The main obstacle to exponential convergence is the existence of slow particles in the system.