Ratner famously classified probability measures invariant under unipotent groups in quotients of Lie groups. For the special case where the unipotent group is a full horosphere such classification is significantly easier and was obtained earlier by Dani and Furstenberg.

Finding analogues in the context of moduli spaces of abelian and quadratic differentials has been a problem that was central in Mirzakhani's research, eventually leading to her remarkable work with Eskin and Mohammadi. As in the homogeneous case, classifying measures invariant under full horospheres is significantly easier also in the moduli spaces context, a classification that was carried out independently by Maryam and I and by U. Hamenstadt.

Ledrappier, Babillot and Sarig discovered that for geometrically infinite surfaces the classification of horospherically invariant locally finite measures is quite different than the geometrically finite case. I will discuss joint work with Or Landesberg that gives new results in this direction.