Phase space contraction of degenerately damped random splittings

When studying out-of-equilibrium systems, one often excites the dynamics in some degrees of freedom while removing the excitation in others through damping. In order for the system to converge to a statistical steady state, the dynamics must transfer the energy from the excited modes to the dissipative directions. The precise mechanisms underlying this transfer are of particular interest and are the topic of this paper. We explore a class of randomly switched models introduced in [2,3] and provide some of the first results showing that minimal damping is sufficient to stabilize the system in a fluids model.