Concavity of the second law of thermodynamics

The second law of thermodynamics states that entropy increases. We interpret this as a physical statement and as an information theoretic statement. We ask to what extent the increase of entropy is built into stochastic processes independently of the physics. Does the increase of entropy provide an arrow of time? What are the assumptions behind Boltzmann's famous H theorem (proving entropy increase) and the quantum version? Do the proofs of the H theorem rely on physical assumptions that imply Markovity or stationarity?.