  In his paper from 1986 Gray developed a theory of dual processes for
attractive spin systems. Based on his work Sturm and Swart systematically
investigated monotonicity-based pathwise dualities for Markov processes in
general and interacting particle systems in particular. In this paper we only
consider monotone interacting particle systems whose state space is the
Cartesian product of countably many copies of a finite set. For this class of
processes Sturm and Swart only showed the well-definedness of their dual
processes if started from finite initial states. This paper closes this gap and
shows how to construct a well-defined pathwise dual process of a monotone
interacting particle system that can also be started from an infinite initial
state. This then allows us to study invariant laws of the dual process and
connect them to properties of the original interacting particle system.
