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Solutions of stationary McKean-Vlasov equation on a high-dimensional sphere and other Riemannian manifolds

Anna Shalova, André Schlichting

TL;DR

This work develops a rigorous framework for stationary McKean–Vlasov equations on compact Riemannian manifolds, with a detailed sphere-specific analysis. It connects stationary states to zeros of a Gibbs map and to critical points of a free-energy functional, establishes existence and (under curvature and kernel-derivative conditions) uniqueness, and analyzes small-noise and high-noise limits. On the sphere, it leverages spherical harmonics and spherical convolution to derive explicit bifurcation curves and phase-transition criteria, including sufficient conditions for discontinuous transitions via resonance. The results are illustrated through applications to noisy transformer models, the Onsager model, and spherical opinion dynamics, highlighting the impact of kernel spectral properties on pattern formation and information retention.

Abstract

We study stationary solutions of McKean-Vlasov equation on a high-dimensional sphere and other compact Riemannian manifolds. We extend the equivalence of the energetic problem formulation to the manifold setting and characterize critical points of the corresponding free energy functional. On a sphere, we employ the properties of spherical convolution to study the bifurcation branches around the uniform state. We also give a sufficient condition for an existence of a discontinuous transition point in terms of the interaction kernel and compare it to the Euclidean setting. We illustrate our results on a range of system, including the particle system arising from the transformer models and the Onsager model of liquid crystals.

Solutions of stationary McKean-Vlasov equation on a high-dimensional sphere and other Riemannian manifolds

TL;DR

This work develops a rigorous framework for stationary McKean–Vlasov equations on compact Riemannian manifolds, with a detailed sphere-specific analysis. It connects stationary states to zeros of a Gibbs map and to critical points of a free-energy functional, establishes existence and (under curvature and kernel-derivative conditions) uniqueness, and analyzes small-noise and high-noise limits. On the sphere, it leverages spherical harmonics and spherical convolution to derive explicit bifurcation curves and phase-transition criteria, including sufficient conditions for discontinuous transitions via resonance. The results are illustrated through applications to noisy transformer models, the Onsager model, and spherical opinion dynamics, highlighting the impact of kernel spectral properties on pattern formation and information retention.

Abstract

We study stationary solutions of McKean-Vlasov equation on a high-dimensional sphere and other compact Riemannian manifolds. We extend the equivalence of the energetic problem formulation to the manifold setting and characterize critical points of the corresponding free energy functional. On a sphere, we employ the properties of spherical convolution to study the bifurcation branches around the uniform state. We also give a sufficient condition for an existence of a discontinuous transition point in terms of the interaction kernel and compare it to the Euclidean setting. We illustrate our results on a range of system, including the particle system arising from the transformer models and the Onsager model of liquid crystals.

Paper Structure

This paper contains 34 sections, 41 theorems, 215 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Compact Riemannian manifold.
  4. Sphere ${\mathbb S}^{n-1}$.
  5. Literature Review
  6. McKean-Vlasov equation as a mean-field limit.
  7. PDEs and free energies on manifolds.
  8. Bifurcations and phase transitions.
  9. Compact Riemannian manifold
  10. Equivalent characterizations of stationary states
  11. Existence of minimizers
  12. Limit of small noise
  13. Geodesic convexity
  14. The structure of the Lebesgue space on the sphere
  15. Spherical harmonics
  16. ...and 19 more sections

Key Result

Theorem 1.1

For any $\gamma \in {\mathbb R}_+$ there exist a solution $\rho_\gamma$ of eq:mckean-vlasov and $\rho_\gamma \in H^1({\mathcal{M}}) \cap {\mathcal{P}}_{ac}({\mathcal{M}})$. In addition, if the curvature of the manifold is bounded from below $\operatorname{Ric}({\mathcal{M}}) \geq \lambda$, $W$ is tw for all $x, y \in {\mathcal{M}}, \ v\in T_x{\mathcal{M}}, u \in T_y{\mathcal{M}}$, then $\rho_\gamm

Theorems & Definitions (80)

  • Theorem 1.1: Existence and uniqueness of solutions
  • Definition 1.2: Spherical harmonics decomposition, see Definition \ref{['def:spherical-decomposition']}
  • Theorem 1.3: Bifurcations
  • Theorem 1.5: Phase transitions
  • Definition 2.1: Weak solution
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Proposition 2.4
  • ...and 70 more