Random Attractors for McKean-Vlasov SDEs

TL;DR

This paper establishes the existence of random attractors for McKean-Vlasov stochastic differential equations on Hilbert spaces, overcoming challenges posed by distribution-dependent coefficients and lack of stochastic flow properties.

Contribution

It develops a general theory for random attractors of cocycles associated with McKean-Vlasov equations on product spaces, including applications to various stochastic PDEs.

Findings

Existence of random attractors for McKean-Vlasov SDEs on Hilbert spaces.

Reduction to singleton attractors with stationary solutions.

Application to stochastic reaction-diffusion and Navier-Stokes equations.

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 $\mathcal{A}(\omega):=(\xi(\omega),\mu_\infty)$, we show that $\xi$ corresponds to the stationary solution for the decoupled SPDE,satisfying $\mathbb{P}\circ[\xi]^{-1}=\mu_\infty$.