Weak Convergence and Linear Systems Duality

For a class of interacting particle systems on a countable set, including the so-called linear systems, self-duality has proved to be a strong tool to show longtime convergence to an invariant state ` from a constant initial state `. This convergence was extended to a class of translation invariant random initial states through the Liggett-Spitzer coupling. Here we drop the assumption of translation invariance of the initial state. Instead, we assume only that there exists a global density ` in a certain L 2 - sense. We use the duality to carry out a comparison argument and show convergence to the same ` as above. We treat some examples in more detail: the parabolic Anderson model, the mutually catalytic branching model, and the smoothing and potlatch processes. 1 Introduction Our purpose here is to present some new weak convergence results for interacting particle systems that enjoy the "linear systems" type duality as described in Chapter IX of Liggett (1985). This particular f..