| Abstract: |
| In this talk, we focus on the existence of random attractors for McKean-Vlasov SDEs on a separable Hilbert space $H$. A significant challenge arises from the distribution-dependence of the coefficients, causing the lack of the stochastic flow property on $H$. To address this, 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, 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. When the attractor reduces to a singleton $\mathcal{A}(\omega):=(\xi(\omega),\mu_{\infty})$, we show $\xi$ is the stationary solution for the decoupled stochastic partial differential equation, satisfying $\mathbb{P}\circ[\xi]^{-1}=\mu_{\infty}$. This talk is based on a joint work with Xianjin Cheng and Zhenxin Liu. |
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