Deterministic--Distance Couplings of Brownian Motions on Radially Isoparametric Manifolds

TL;DR

This paper introduces a geometric framework for coadapted Brownian couplings on radially isoparametric manifolds, deriving bounds on inter-particle distance evolution and extending curvature classification to all such manifolds.

Contribution

It provides a unified geometric approach to characterize Brownian couplings on RIM, extending prior constant-curvature results to variable curvature settings, and links curvature data with stochastic coupling dynamics.

Findings

Derived an intrinsic drift-window inequality for inter-particle distance

Proved the bound is necessary and sufficient for prescribed distance laws

Connected extremal couplings with geometric curvature data

Abstract

We develop a unified geometric framework for coadapted Brownian couplings on radially isoparametric manifolds (RIM)--spaces whose geodesic spheres have principal curvatures $\kappa_1(r),\dots,\kappa_{n-1}(r)$ depending only on the geodesic radius $r$. The mean curvature of such a geodesic sphere is denoted by $A(r) = \mathrm{Tr}(S_r) = \sum_{i=1}^{n-1} \kappa_i(r)$, where $S_r$ is the shape operator of the sphere of radius $r$. Within the stochastic two--point It\^{o} formalism, we derive an intrinsic drift--window inequality \[ A(r) - \sum_i |\kappa_i(r)| \;\le\; \rho'(t) \;\le\; A(r) + \sum_i |\kappa_i(r)|, \] governing the deterministic evolution of the inter--particle distance $\rho_t = d(X_t, Y_t)$ 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 $\rho(t)$, 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.