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A note on the Long-Time behaviour of Stochastic McKean-Vlasov Equations with common noise

Raphael Maillet

TL;DR

The paper investigates the long-time behavior of stochastic McKean–Vlasov equations with common noise, focusing on invariant measures for the measure-valued flow $\{m_t\}$ when the drift is linear in the measure via $b(x,m)=-\nabla V(x)-\nabla W\!*m(x)$. It proves existence of an invariant measure under mild assumptions, and, in the case of a uniformly convex confinement $V$, establishes uniform-in-time propagation of chaos and uniqueness of the invariant measure with exponential convergence; in the non-convex setting without idiosyncratic noise ($\sigma=0$), common noise restores uniqueness and yields exponential convergence, with an explicit invariant measure in the quadratic interaction scenario. The analysis combines mean-field limits, coupling methods (including synchronous and reflection couplings), and leverages deformation metrics on probability measures to handle the infinite-dimensional state space. These results highlight a stabilizing role for common noise in systems that would otherwise admit multiple equilibria, and provide concrete rates of convergence toward equilibrium. Overall, the work advances understanding of asymptotic stability for measure-valued stochastic dynamics with common noise and delineates regimes where uniqueness and rapid convergence can be guaranteed.

Abstract

This paper focuses on the long-term behavior of solutions to nonlinear stochastic Fokker-Planck equations driven by common noise, where the drift term has a linear dependence on the measure. These equations, which describe the evolution of probability distributions, naturally arise in the mean-field limit of interacting particle systems driven by both idiosyncratic and common noises. After proving the existence of an invariant measure under some mild conditions, we first consider the case where the confinement potential is uniformly convex. In this setting, we establish a result of uniform-in-time conditional propagation of chaos for the associated particle system. This result directly implies the uniqueness of the long-term behavior for solutions of the nonlinear stochastic Fokker-Planck equation. Then, we highlight a more surprising phenomenon of uniqueness recovery induced by the addition of common noise in the non-convex case, albeit under more restrictive structural assumptions. Specifically, we show that the presence of common noise leads to uniqueness and exponential convergence towards equilibrium in the absence of idiosyncratic noise. This result emphasizes the stabilizing role of common noise in systems where non-convex potentials would typically allow for multiple invariant measures.

A note on the Long-Time behaviour of Stochastic McKean-Vlasov Equations with common noise

TL;DR

The paper investigates the long-time behavior of stochastic McKean–Vlasov equations with common noise, focusing on invariant measures for the measure-valued flow when the drift is linear in the measure via . It proves existence of an invariant measure under mild assumptions, and, in the case of a uniformly convex confinement , establishes uniform-in-time propagation of chaos and uniqueness of the invariant measure with exponential convergence; in the non-convex setting without idiosyncratic noise (), common noise restores uniqueness and yields exponential convergence, with an explicit invariant measure in the quadratic interaction scenario. The analysis combines mean-field limits, coupling methods (including synchronous and reflection couplings), and leverages deformation metrics on probability measures to handle the infinite-dimensional state space. These results highlight a stabilizing role for common noise in systems that would otherwise admit multiple equilibria, and provide concrete rates of convergence toward equilibrium. Overall, the work advances understanding of asymptotic stability for measure-valued stochastic dynamics with common noise and delineates regimes where uniqueness and rapid convergence can be guaranteed.

Abstract

This paper focuses on the long-term behavior of solutions to nonlinear stochastic Fokker-Planck equations driven by common noise, where the drift term has a linear dependence on the measure. These equations, which describe the evolution of probability distributions, naturally arise in the mean-field limit of interacting particle systems driven by both idiosyncratic and common noises. After proving the existence of an invariant measure under some mild conditions, we first consider the case where the confinement potential is uniformly convex. In this setting, we establish a result of uniform-in-time conditional propagation of chaos for the associated particle system. This result directly implies the uniqueness of the long-term behavior for solutions of the nonlinear stochastic Fokker-Planck equation. Then, we highlight a more surprising phenomenon of uniqueness recovery induced by the addition of common noise in the non-convex case, albeit under more restrictive structural assumptions. Specifically, we show that the presence of common noise leads to uniqueness and exponential convergence towards equilibrium in the absence of idiosyncratic noise. This result emphasizes the stabilizing role of common noise in systems where non-convex potentials would typically allow for multiple invariant measures.
Paper Structure (16 sections, 13 theorems, 139 equations)

This paper contains 16 sections, 13 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. The process $m$ as a mean field limit
  3. Existence of an invariant measure for the stochastic flow of measures.
  4. The existence result
  5. The case of a uniformly convex confinement potential
  6. Uniform in time propagation of chaos
  7. Uniqueness of the invariant measure
  8. Discussion and example
  9. Non-convex potential without idiosyncratic noise
  10. Existence of an invariant measure
  11. Uniqueness of the invariant measure and exponential decay
  12. Discussion and Heuristics
  13. Proof of Theorem \ref{['th:sg0']}
  14. Proof of Proposition \ref{['Lacker']}
  15. Proof of Lemma \ref{['le:L2']}
  16. ...and 1 more sections

Key Result

Lemma 1

The measure-valued process $(m_t)_{t\geq 0}$ is a solution in the weak sense of the following Stochastic Partial Differential Equation: with initial condition $\mathcal{L}\left( m_0\right) = P_0$.

Theorems & Definitions (29)

  • Lemma 1
  • Definition 1
  • Definition 2: Invariant measure
  • Definition 3
  • Remark 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Definition 4
  • ...and 19 more