Expanding Measures: Random Walks and Rigidity on Homogeneous Spaces

The paper’s Theorem 1.1 states that for an H-expanding measure µ with finite first moment, any µ-ergodic µ-stationary probability measure ν on X = G/Λ is Γµ-invariant and homogeneous, and that StabG(ν)° is normalized by H (and in fact equals the connected component of the homogeneous acting subgroup arising in the proof). The proof proceeds via Eskin–Lindenstrauss’ classification (Theorem 4.2), after establishing uniform expansion modulo the centralizer using Lemma 4.3; the argument then shows homogeneity and identifies the connected stabilizer as N° in the homogeneous description (see Theorem 1.1, Lemma 4.3, Theorem 4.2, and the end of the proof of Theorem 1.1). The candidate solution outlines the same route: it uses H-expansion (in particular Ad-expansion), the finite-moment hypothesis, applies the Benoist–Quint/Eskin–Lindenstrauss classification framework (also via Simmons–Weiss and Prohaska–Sert–Shi), concludes homogeneity and Γµ-invariance in the ergodic case, and notes that the connected stabilizer is normalized by H. Minor imprecision: the candidate’s Step 1 asserts homogeneity for every stationary (not necessarily ergodic) measure; the paper states and proves the result for µ-ergodic measures, with the non-ergodic case handled by ergodic decomposition. Aside from that mild overreach, the approaches match closely. Citations: Theorem 1.1 (paper’s main claim) ; Eskin–Lindenstrauss alternative (Theorem 4.2) and its use in the proof ; the H-expansion to “uniformly expanding mod centralizer” step (Lemma 4.3) ; identification of the connected stabilizer with N° in the proof of Theorem 1.1 .