Classification of extremal invariant measures

Determine all extremal invariant measures for zero-range and exclusion processes across all model settings, including settings beyond the dimensions, rate functions, and transition kernels for which classification is currently known.

Background

The notes prove extremality of the translation-invariant product invariant measures $\nu_\rho$ for a class of infinite-volume zero-range processes. They also mention that in dimensions one and two, under increasing-rate assumptions, known results identify all extremals in certain translation-invariant and non-translation-invariant settings.

A complete classification for zero-range and exclusion processes in all model settings is not available. The unresolved issue concerns identifying every extremal invariant measure, rather than merely constructing or proving extremality of a distinguished product family.