On the Uniqueness and Nonuniqueness of Proximity Processes

Abstract

We discuss the uniqueness of a class of infinite particle systems known as proximity processes, with the aid of certain "dual" Markov chains. By checking whether the dual "explodes," i.e., attempts infinitely many jumps in a finite time, and how it explodes when it does, it is possible in many cases to determine whether or not there is more than one particle system with given flip rates. We then use duality to find an example of a system which is uniquely determined by its flip rates, but whose generator is not the closure of the naive operator formed from these flip rates.