On the Existence of the Dual Right Markov Process and Applications

Abstract

We show that given a Borel right process there exists a dual process which is also a right Markov process. However, it is necessary to enlarge the initial space to a new Lusin topological space and the dual process is a right process with respect to a second Lusin topology. As a result both processes can be identified as solutions to martingale problems. Another application is the proof that the Riesz decomposition holds (in potential and harmonic components) for the excessive functions and the set of all potentials becomes solid in order.