報告人：胡杉杉 博士

報告題目：Random dynamical systems for McKean-Vlasov SDEs via rough path theory

報告時間：2025年10月11日（周六）下午3：00

報告地點：云龍校區6號樓304會議室

主辦單位：數學與統計學院、數學研究院、科學技術研究院

報告人簡介：

胡杉杉，天津大學和柏林工業大學在讀博士生，師從王鳳雨教授和Benjamin Gess教授，主要研究方向為隨機微分方程和隨機動力系統，目前已在Annals of Applied Probability上發表論文。

報告摘要：

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 to non-linear Fokker--Planck equations associated to the EKS is shown.