Random dynamical systems for McKean--Vlasov SDEs via rough path theory

TL;DR

This paper establishes the existence of random dynamical systems for McKean--Vlasov SDEs using rough path theory, addressing both SDE and PDE aspects, with applications to ensemble Kalman sampler and Landau equation.

Contribution

It introduces a novel approach combining rough path theory and PDE analysis to construct RDS for McKean--Vlasov SDEs, including degenerate cases.

Findings

Proves existence of RDS for McKean--Vlasov SDEs.

Develops a pathwise solution theory for SDEs with time-dependent coefficients.

Shows uniqueness of solutions for nonlinear Fokker--Planck equations in EKS context.

Abstract

The existence of random dynamical systems for McKean--Vlasov SDEs is established. This is approached by considering the joint dynamics of the corresponding nonlinear Fokker-Planck equation governing the law of the system and the underlying stochastic differential equation (SDE) as a dynamical system on the product space $\RR^d \times \mathcal{P}(\RR^d)$. The proof relies on two main ingredients: At the level of the SDE, a pathwise rough path-based solution theory for SDEs with time-dependent coefficients is implemented, while at the level of the PDE a well-posedness theory is developed, for measurable solutions and allowing for degenerate diffusion coefficients. The results apply in particular to the so-called ensemble Kalman sampler (EKS), proving the existence of an associated RDS under some assumptions on the posterior, as well as to the Lagrangian formulation of the Landau equation with Maxwell molecules. As a by-product of the main results, the uniqueness of solutions non-linear Fokker--Planck equations associated to the EKS is shown.