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Random Attractors for McKean-Vlasov SDEs

Mengyu Cheng, Xianjin Cheng, Zhenxin Liu

TL;DR

This work addresses the long-time behavior of McKean–Vlasov stochastic dynamics by establishing random attractors for SDEs and SPDEs on a Hilbert space. It develops a two-parameter random dynamical systems framework on the product space $H\times\mathcal{P}_2(H)$ via a skew-product/cocycle approach, enabling pullback attractors despite distribution-dependent coefficients. The authors prove general attractor results and apply them to MV SODE, MV stochastic reaction-diffusion, and MV stochastic 2D Navier–Stokes equations, showing the existence of random attractors and, in certain cases, singleton attractors corresponding to stationary solutions with invariant measures $\mu_{\infty}$. These results provide pathwise descriptions of asymptotic behavior for mean-field stochastic systems and offer rigorous attractor structures for complex MV dynamics.

Abstract

In this paper, we mainly focus on the existence of random attractors for McKean-Vlasov stochastic differential equations on a separable Hilbert space $H$. A significant challenge arises from the distribution-dependence of the coefficients, thereby causing the lack of the stochastic flow property on $H$. To address this issue, we first transform the original equation into a system on the product space $H \times \mathcal{P}(H)$ and consider the existence of random attractors on this space. We then analyze cocycles associated with two parametric dynamical systems. Within this framework, we define the corresponding pullback random attractor and develop a general theory for the existence of random attractors for such cocycles. Finally, we apply our theoretical results to McKean-Vlasov stochastic ordinary differential equations, McKean-Vlasov stochastic reaction-diffusion equations, and McKean-Vlasov stochastic 2D Navier-Stokes equations. In the case where the attractor reduces to a singleton set $\mathcal{A}(ω):=(ξ(ω),μ_\infty)$, we show that $ξ$ corresponds to the stationary solution for the decoupled SPDE,satisfying $\mathbb{P}\circ[ξ]^{-1}=μ_\infty$.

Random Attractors for McKean-Vlasov SDEs

TL;DR

This work addresses the long-time behavior of McKean–Vlasov stochastic dynamics by establishing random attractors for SDEs and SPDEs on a Hilbert space. It develops a two-parameter random dynamical systems framework on the product space via a skew-product/cocycle approach, enabling pullback attractors despite distribution-dependent coefficients. The authors prove general attractor results and apply them to MV SODE, MV stochastic reaction-diffusion, and MV stochastic 2D Navier–Stokes equations, showing the existence of random attractors and, in certain cases, singleton attractors corresponding to stationary solutions with invariant measures . These results provide pathwise descriptions of asymptotic behavior for mean-field stochastic systems and offer rigorous attractor structures for complex MV dynamics.

Abstract

In this paper, we mainly focus on the existence of random attractors for McKean-Vlasov stochastic differential equations on a separable Hilbert space . A significant challenge arises from the distribution-dependence of the coefficients, thereby causing the lack of the stochastic flow property on . To address this issue, we first transform the original equation into a system on the product space and consider the existence of random attractors on this space. We then analyze cocycles associated with two parametric dynamical systems. Within this framework, we define the corresponding pullback random attractor and develop a general theory for the existence of random attractors for such cocycles. Finally, we apply our theoretical results to McKean-Vlasov stochastic ordinary differential equations, McKean-Vlasov stochastic reaction-diffusion equations, and McKean-Vlasov stochastic 2D Navier-Stokes equations. In the case where the attractor reduces to a singleton set , we show that corresponds to the stationary solution for the decoupled SPDE,satisfying .

Paper Structure

This paper contains 14 sections, 34 theorems, 319 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. General results
  4. Random attractors for McKean-Vlasov SODE
  5. Random attractors for McKean-Vlasov SPDE
  6. McKean-Vlasov stochastic reaction-diffusion equation
  7. McKean-Vlasov stochastic 2D Navier-Stokes equation
  8. Proof of Proposition \ref{['prop0117']} and Theorem \ref{['thm0330']}
  9. Proof of main results for McKean-Vlasov SODE
  10. Proof of Theorem \ref{['thmSODE']}
  11. Proof of Example \ref{['ex01']}
  12. Proof of main results for McKean-Vlasov SPDE
  13. McKean-Vlasov stochastic reaction-diffusion equation
  14. McKean-Vlasov stochastic 2D Navier-Stokes equation

Key Result

Proposition 2.2

Assume that $\{\theta_{2,t}\}_{t\geq0}$ is continuous over any $B_{{\mathcal{P}}}\in{\mathscr S}$, and admits a compact absorbing set $B\in{\mathscr S}$. Then $\{\theta_{2,t}\}_{t\geq0}$ admits a ${\mathscr S}$-global attractor ${\mathcal{P}}^*$. Suppose further that ${\mathcal{P}}$ is a Polish spac Then $\Phi$ admits a ${\mathscr D}$-random attractor ${\mathscr A}$, which is defined by where

Theorems & Definitions (84)

  • Definition 2.1
  • Proposition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • Theorem 2.8
  • Remark 2.9
  • Remark 2.10
  • ...and 74 more