2025/11/06 by Gunhee Cho, Hyun Chul Jang, Cho, Gunhee +3
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Point processes and geometric inequalities #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2511.04431
openalex publication_date 2025/11/06 · openalex created_date 2025/11/08 · openalex updated_date 2026/07/28
We develop a unified geometric framework for coadapted Brownian couplings on radially isoparametric manifolds (RIM)--spaces whose geodesic spheres have principal curvatures κ1(r),…,κn-1(r) depending only on the geodesic radius r. The mean curvature of such a geodesic sphere is denoted by A(r) = Tr(Sr) = ∑i=1n-1 κi(r), where Sr is the shape operator of the sphere of radius r. Within the stochastic two--point Itô formalism, we derive an intrinsic drift--window inequality A(r) - ∑i |κi(r)| ≤ ρ'(t) ≤ A(r) + ∑i |κi(r)|, governing the deterministic evolution of the inter--particle distance ρt = d(Xt, Yt) under all coadapted couplings. We prove that this bound is both necessary and sufficient for the existence of a coupling realizing any prescribed distance law ρ(t), thereby extending the constant--curvature classification of Pascu--Popescu (2018) to all RIM. The endpoints of the drift window correspond to the synchronous and reflection couplings, providing geometric realizations of extremal stochastic drifts. Applications include stationary fixed--distance couplings on compact--type manifolds, linear escape laws on asymptotically hyperbolic spaces, and rigidity of rank--one symmetric geometries saturating the endpoint bounds. This establishes a direct correspondence between radial curvature data and stochastic coupling dynamics, linking Riccati comparison geometry with probabilistic coupling theory.