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Connecting Orbits in Cooperative McKean-Vlasov SDEs

Chunlin Liu, Baoyou Qu, Jinxiang Yao, Yanpeng Zhi

TL;DR

This work develops a monotone dynamical systems approach for cooperative McKean–Vlasov SDEs with multiplicative noise, establishing a framework to study non-uniqueness of invariant measures and their connecting orbits in Wasserstein space under the stochastic order. By extending the state space to finite signed measures with a compatible cone and adapting the Dancer–Hess connecting orbit theorem to this setting, the authors prove the existence of multiple order-related invariant measures and monotone connecting orbits, along with shrinking neighborhoods that clarify instability directions. The results apply to classical MV-SDE models, including granular media in double-well and multi-well landscapes and two-species population dynamics, and provide explicit parameter regimes and phase diagrams. The methodology offers a global dynamical perspective on phase transitions and metastability in interacting diffusions, linking local dissipativity to global order-driven transitions.

Abstract

In this work we extend the framework of monotone dynamical systems to a broad and important class of stochastic equations, namely cooperative McKean-Vlasov SDEs with multiplicative noise. Under a locally dissipative assumption, our main theorem establishes the existence of multiple order-related invariant measures in the the Wasserstein space together with monotone connecting orbits (heteroclinic orbits) between them, with respect to the stochastic order. The presence of such connecting orbits also reveals the unstable nature of those invariant measures appearing as their backward limits, a dynamical feature that has remained largely unexplored in stochastic equations. The framework applies to a wide range of classical models, including granular media equations in double-well and multi-well confining potentials with quadratic interaction, perturbed double-well landscapes, and interacting multi-species population models. Our method is based on building a monotone dynamical system that preserves the stochastic order, achieved through a cone compatible with this order and an extension of the classical Dancer-Hess connecting orbit theorem.

Connecting Orbits in Cooperative McKean-Vlasov SDEs

TL;DR

This work develops a monotone dynamical systems approach for cooperative McKean–Vlasov SDEs with multiplicative noise, establishing a framework to study non-uniqueness of invariant measures and their connecting orbits in Wasserstein space under the stochastic order. By extending the state space to finite signed measures with a compatible cone and adapting the Dancer–Hess connecting orbit theorem to this setting, the authors prove the existence of multiple order-related invariant measures and monotone connecting orbits, along with shrinking neighborhoods that clarify instability directions. The results apply to classical MV-SDE models, including granular media in double-well and multi-well landscapes and two-species population dynamics, and provide explicit parameter regimes and phase diagrams. The methodology offers a global dynamical perspective on phase transitions and metastability in interacting diffusions, linking local dissipativity to global order-driven transitions.

Abstract

In this work we extend the framework of monotone dynamical systems to a broad and important class of stochastic equations, namely cooperative McKean-Vlasov SDEs with multiplicative noise. Under a locally dissipative assumption, our main theorem establishes the existence of multiple order-related invariant measures in the the Wasserstein space together with monotone connecting orbits (heteroclinic orbits) between them, with respect to the stochastic order. The presence of such connecting orbits also reveals the unstable nature of those invariant measures appearing as their backward limits, a dynamical feature that has remained largely unexplored in stochastic equations. The framework applies to a wide range of classical models, including granular media equations in double-well and multi-well confining potentials with quadratic interaction, perturbed double-well landscapes, and interacting multi-species population models. Our method is based on building a monotone dynamical system that preserves the stochastic order, achieved through a cone compatible with this order and an extension of the classical Dancer-Hess connecting orbit theorem.
Paper Structure (17 sections, 27 theorems, 243 equations, 1 figure)

This paper contains 17 sections, 27 theorems, 243 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notations
  3. Main results
  4. Stochastic Order and Compatible Cone
  5. Properties of stochastic order
  6. Construction of compatible cone
  7. Generation of Monotone Dynamical Systems
  8. Joint continuity of semigroups of McKean-Vlasov SDEs
  9. Comparison principle with respect to stochastic order
  10. Connecting Orbit Theorem in Monotone Dynamical Systems
  11. Preliminaries of dynamical systems on ordered spaces
  12. Statement and proof of connecting orbit theorem
  13. Order-Related Invariant Measures with Shrinking Neighbourhoods
  14. Properties of measure-iterating map
  15. Existence of order-related invariant measures
  16. ...and 2 more sections

Key Result

Theorem 1.1

Suppose that Assumption asp:lipschitz, asp:cooperation, asp:dissipative-growth-nondegeneracy hold, and there exists $\{a_i\}_{i=1}^n\subset \mathbb R^d$ for some $n\geq 2$ such that the equation eq:mvsystem is locally dissipative at $a_i$ with configurations $(r_{a_i},g_{a_i})$ for all $1\leq i\leq then the equation eq:mvsystem has at least $(2n-1)$ order-related invariant measures in $\mathcal{P

Figures (1)

  • Figure 1: Phase diagram for double-well landscapes

Theorems & Definitions (66)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2: Double-well landscapes, Figure \ref{['fig:double-well']}
  • Remark 1.2
  • Theorem 1.3: Multi-well landscapes
  • Theorem 1.4: Double-well with perturbation
  • Theorem 1.5: Multi-species population model
  • Definition 2.1
  • ...and 56 more