Polar sets and Palm measures in the theory of flows

Abstract:

Given a flow $({\theta _t}),t$ real, over a probability space $\Omega$, we prove that certain measures on $\Omega$ (viewed as the state space of the flow) decompose uniquely into a Palm measure $Q$ which charges no “polar set” and a measure supported by a polar set. Considering the continuous and discrete parts of the additive functional corresponding to $Q$, we find that $Q$ further decomposes into a measure charging no “semipolar set” and a measure supported by one. As a consequence, Palm measures are exactly those which neglect sets which the flow neglects, and polar sets are exactly those neglected by every Palm measure. Finally, we characterize various properties, such as predictability and continuity, of an additive functional in terms of its Palm measure. These results further illuminate the role played by supermartingales in the theory of flows, as pointed by J. de Sam Lazaro and P. A. Meyer.