Stationary product measures for conservative particle systems and ergodicity criteria

Abstract

We study conservative particle systems on $W^S$, where $S$ is countable and $W = \{0, \dots, N\}$ or $W = \mathbb{N}$, where the generator reads

\[Lf(\eta) = \sum_{x,y} p(x,y) b(\eta_x,\eta_y) (f(\eta - \delta_x + \delta_y) - f(\eta)).\]

Under assumptions on $b$ and the assumption that $p$ is finite range, which allow for the exclusion, zero range and misanthrope processes, we determine exactly what the stationary product measures are.

Furthermore, under the condition that $p + p^*$, $p^*(x,y) := p(y,x)$, is irreducible, we show that a stationary measure $\mu$ is ergodic if and only if the tail sigma algebra of the partial sums is trivial under $\mu$. This is a consequence of a more general result on interacting particle systems that shows that a stationary measure is ergodic if and only if the sigma algebra of sets invariant under the transformations of the process is trivial. We apply this result combined with a coupling argument to the stationary product measures to determine which product measures are ergodic. For the case that $W$ is finite, this gives a complete characterisation.

In the case that $W = \mathbb{N}$, it holds for nearly all functions $b$ that a stationary product measure is ergodic if and only if it is supported by configurations with an infinite amount of particles. We show that this picture is not complete. We give an example of a system where $b$ is such that there is a stationary product measure which is not ergodic, even though it concentrates on configurations with an infinite number of particles.