Duality theory for Markov processes: Part 1

TL;DR

This paper develops the foundational theory of duality for Markov processes, proving key existence theorems and establishing correspondences between measures, thereby advancing the mathematical understanding of Markov process duality.

Contribution

It provides the first comprehensive proof of Fitzsimmons' existence theorem for moderate Markov dual processes and links random measures with sigma finite measures, laying groundwork for future duality studies.

Findings

Proved Fitzsimmons' existence theorem for dual processes

Established correspondence between random measures and sigma finite measures

Provided detailed proofs of duality-related results

Abstract

This is the first part of a possible monograph on the duality of Markov processes. It contains a proof of Fitzsimmons' existence theorem of a moderate Markov dual process relative to an excessive measure, m, together with the necessary preliminary material. Then this is applied to prove the correspondence between optional copredictable homogenous random measures and sigma finite measures not charging m-exceptional sets again following Fitzsimmons. The second part which may never be written would deal with duality proper including results from, but not limited to, my joint paper with P. J. Fitzsimmons"Potential Theory of Moderate Markov Dual Processes" which appeared in Potential Anal.(2009) 31:275-310. Complete proofs of all results not appearing in standard books are given with the one exception of Dellacherie's result characterizing semipolar sets.