Spontaneous Inflation and the Origin of the Arrow of Time
  We suggest that spontaneous eternal inflation can provide a natural explanation for the thermodynamic arrow of time, and discuss the underlying assumptions and consequences of this view. In the absence of inflation, we argue that systems coupled to gravity usually evolve asymptotically to the vacuum, which is the only natural state in a thermodynamic sense. In the presence of a small positive vacuum energy and an appropriate inflaton field, the de Sitter vacuum is unstable to the spontaneous onset of inflation at a higher energy scale. Starting from de Sitter, inflation can increase the total entropy of the universe without bound, creating universes similar to ours in the process. An important consequence of this picture is that inflation occurs asymptotically both forwards and backwards in time, implying a universe that is (statistically) time-symmetric on ultra-large scales. 
Sean M. Carroll and Jennifer Chen