Pathwise duals of monotone and additive Markov processes

TL;DR

This paper systematically develops pathwise dualities for Markov processes in partially ordered sets, unifying and extending known dualities, and generalizing graphical representations to broader classes of processes.

Contribution

It introduces a unified framework for dualities of Markov processes in lattices, extending existing dualities and providing new constructions, especially for additive and monotone processes.

Findings

Unified treatment of several classical dualities.

Generalization of graphical representations to additive Markov processes.

Extension of duality concepts to processes on general lattices.

Abstract

This paper develops a systematic treatment of monotonicity-based pathwise dualities for Markov processes taking values in partially ordered sets. We show that every Markov process that takes values in a finite partially ordered set and whose generator can be represented in monotone maps has a pathwise dual process. In the special setting of attractive spin systems this has been discovered earlier by Gray. We show that the dual simplifies a lot when the state space is a lattice (in the order-theoretic meaning of the word) and all monotone maps satisfy an additivity condition. This leads to a unified treatment of several well-known dualities, including Siegmund's dual for processes with a totally ordered state space, duality of additive spin systems, and a duality due to Krone for the two-stage contact process, and allows for the construction of new dualities as well. We show that the well-known representation of additive spin systems in terms of open paths in a graphical representation can be generalized to additive Markov processes taking values in general lattices, but for the process and its dual to be representable on the same underlying space, we need to assume that the lattice is distributive. In the final section, we show how our results can be generalized from finite state spaces to interacting particle systems with finite local state spaces.