Equilibration and asymptotic stationarity for non-Markovian master equations

We show that certain positivity-preserving non-Markovian generalizations of the Kossakowski-Lindblad master equation can exhibit equilibration to an asymptotic state which is stationary with respect to the shifted system Hamiltonian for general system-bath coupling. This is in sharp contrast to results for Markovian forms which require strong relations between these operators (e.g. commutation of isolated system Hamiltonian and coupling operator). We also expand the list of sufficient conditions for positivity of non-Markovian master equations.