Uniqueness of equivariant singular Bott-Chern classes

In this paper, we shall discuss possible theories of defining equivariant singular Bott-Chern classes and corresponding uniqueness property. By adding a natural axiomatic characterization to the usual ones of equivariant Bott-Chern secondary characteristic classes, we will see that the construction of Bismut's equivariant Bott-Chern singular currents provides a unique way to define a theory of equivariant singular Bott-Chern classes. This generalizes J. I. Burgos Gil and R. Li\c{t}canu's discussion to the equivariant case. As a byproduct of this study, we shall prove a concentration formula which can be used to prove an arithmetic concentration theorem in Arakelov geometry.