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

Gunhee Cho, Hyun Chul Jang, Taeik Kim

TL;DR

This work extends the deterministic-distance coupling framework from constant-curvature spaces to the broader class of radially isoparametric manifolds (RIM). It derives a sharp drift window $A(r) \,\pm\,\sum_i|\kappa_i(r)|$ for the distance between two coadapted Brownian particles, proving this window is both necessary and sufficient for realizing any prescribed distance law, with the endpoints attained by synchronous and reflection couplings. The authors develop a full geometric–stochastic machinery via two-point Itô calculus to connect radial curvature data to coupling dynamics, and they classify static and dynamic regimes, yielding fixed-distance realizations and asymptotic linear speeds characterized by the limits $A_\infty$ and $\kappa_{\infty,i}$. Applications include fixed-distance couplings on compact-type manifolds and linear-escape laws on asymptotically hyperbolic spaces, plus rigidity results for rank-one symmetric geometries saturating the endpoint bounds. The work thus unifies Riccati comparison geometry with stochastic coupling theory in a broad geometric setting, extending classical results from space forms to the full RIM class.

Abstract

We develop a unified geometric framework for coadapted Brownian couplings on radially isoparametric manifolds (RIM)--spaces whose geodesic spheres have principal curvatures $κ_1(r),\dots,κ_{n-1}(r)$ depending only on the geodesic radius $r$. The mean curvature of such a geodesic sphere is denoted by $A(r) = \mathrm{Tr}(S_r) = \sum_{i=1}^{n-1} κ_i(r)$, where $S_r$ is the shape operator of the sphere of radius $r$. Within the stochastic two--point Itô formalism, we derive an intrinsic drift--window inequality \[ A(r) - \sum_i |κ_i(r)| \;\le\; ρ'(t) \;\le\; A(r) + \sum_i |κ_i(r)|, \] governing the deterministic evolution of the inter--particle distance $ρ_t = d(X_t, Y_t)$ 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 $ρ(t)$, 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.

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

TL;DR

This work extends the deterministic-distance coupling framework from constant-curvature spaces to the broader class of radially isoparametric manifolds (RIM). It derives a sharp drift window for the distance between two coadapted Brownian particles, proving this window is both necessary and sufficient for realizing any prescribed distance law, with the endpoints attained by synchronous and reflection couplings. The authors develop a full geometric–stochastic machinery via two-point Itô calculus to connect radial curvature data to coupling dynamics, and they classify static and dynamic regimes, yielding fixed-distance realizations and asymptotic linear speeds characterized by the limits and . Applications include fixed-distance couplings on compact-type manifolds and linear-escape laws on asymptotically hyperbolic spaces, plus rigidity results for rank-one symmetric geometries saturating the endpoint bounds. The work thus unifies Riccati comparison geometry with stochastic coupling theory in a broad geometric setting, extending classical results from space forms to the full RIM class.

Abstract

We develop a unified geometric framework for coadapted Brownian couplings on radially isoparametric manifolds (RIM)--spaces whose geodesic spheres have principal curvatures depending only on the geodesic radius . The mean curvature of such a geodesic sphere is denoted by , where is the shape operator of the sphere of radius . Within the stochastic two--point Itô formalism, we derive an intrinsic drift--window inequality governing the deterministic evolution of the inter--particle distance 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 , 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.

Paper Structure

This paper contains 24 sections, 32 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Geometric framework
  3. Notation and conventions.
  4. Mixed Hessian and the distance function
  5. Definitions and hierarchy
  6. General radial formulas and specializations
  7. Stochastic preliminaries and coadapted couplings
  8. Common notation and standing assumptions
  9. Brownian motion via stochastic development
  10. Coadapted couplings of Brownian motions
  11. Product diffusion and its time-dependent generator
  12. Generator on the distance and cut locus handling
  13. Canonical examples and extremality
  14. Distance SDE and sharp drift window
  15. Two–point Itô formula for the distance
  16. ...and 9 more sections

Key Result

Proposition 2.1

Let $\rho(x,y)=d(x,y)$ be smooth on $\mathcal{U}$. Fix $(x,y)\in\mathcal{U}$ with $r=d(x,y)$ and let $\gamma:[0,r]\to M$ be the unique minimizing unit-speed geodesic from $x$ to $y$. For $X\in T_xM$, let $\tilde{X}=\mathcal{P}_{x\to y}X\in T_yM$. Writing $X_T$ for the orthogonal projection of $X$ on where $S_r$ is the shape operator of $\mathbb{S}_r(x)$ with outward normal $-\nabla_x\rho$.

Theorems & Definitions (72)

  • Proposition 2.1: Mixed Hessian of the distance
  • proof
  • Remark 2.2: Rotationally symmetric model
  • Definition 2.3: Radially isoparametric manifold Sakai1996, Petersen2016
  • Definition 2.4: Rotationally symmetric manifold Sakai1996, Petersen2016
  • Definition 2.5: Asymptotically hyperbolic manifold GrahamLee1991, MazzeoMelrose1987
  • Definition 2.6: Asymptotically isoparametric manifold Helgason2001
  • Remark 2.7: Class inclusions and examples
  • Proposition 2.8: Second fundamental form and Laplacian of $r$
  • Lemma 2.1: Riccati equation along radial geodesics Petersen2016
  • ...and 62 more