Characterizing measures according to their Radon–Nikodym cocycles: canonical marked Gibbs measures under the action of the diffeomorphism group - Kuna, Tobias; Goldin, Gerald A.; Kondratiev, Yuri G.; Silva, José L. (Ukrains’kyi Matematychnyi Zhurnal, 2025)
UDC 517.98 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.