Characterizing Measures According to Their Radon–Nikodym Cocycles: Canonical Marked Gibbs Measures Under the Action of the Diffeomorphism Group

Suppose that we have a canonical Gibbs measure μ defined on a marked configuration space Ω that describes a system of infinitely many indistinguishable particles with internal degrees of freedom together with a diffeomorphism group action on Ω. Then μ is quasiinvariant under the group action, and we obtain a class of associated cocycles from its Radon–Nikodym derivatives. The cocycles are defined up to μ-measure zero sets. We show that it is possible to choose a suitable pointwise-defined version β of this cocycle. Further, we characterize all the measures on Ω that possess β as their cocycle. If μ is obtained (e.g.) from a particular two-body potential V^ (satisfying some mild regularity assumptions), then β takes a certain explicit form, and the class of canonical Gibbs measures characterized by β contains exactly the measures associated with the potential V^. Our result is based on the inheritance properties for the characterization by cocycles of Radon–Nikodym derivatives, which are proved for general G-spaces for local infinite-dimensional groups.