CS46: Asymptotic behaviors for McKean-Vlasov Stochastic Differential Equations
date: 7/17/2025, time: 16:00-17:30, room: IM 605
Organizer: Huijie Qiao (Southeast University)
Chair: Huijie Qiao (Southeast University)
Averaging principles and central limit theorems for multiscale McKean-Vlasov stochastic systems
Huijie Qiao (Southeast University)
In this paper, we study a class of multiscale McKean-Vlasov stochastic systems where the entire system depends on the distribution of the fast component. First of all, by the Poisson equation method we prove that the slow component converges to the solution of the averaging equation in the \(L^p\) (\(p\geq 2\)) space with the optimal convergence rate \(\frac12\). Then a central limit theorem is established by tightness.
Characterisation of optimal solutions to second-order Beckmann problem through bimartingale couplings and leaf decompositions
Krzysztof Ciosmak (University of Toronto)
I will provide a complete characterisation of the optimal solutions for
the three-marginal optimal transport problem - introduced in \([1]\), and
whose relaxation is the second-order Beckmann problem - for arbitrary
pairs \(\mu,\nu\in\mathcal{P}_2(\mathbb{R}^n)\) of absolutely continuous
measures with common barycentre such that there exists an optimal plan
with absolutely continuous third marginal.
I will define the concept of bimartingale couplings for a pair of
measures and establish several equivalent conditions that ensure such
couplings exist. One of these conditions is that the pair is ordered
according to the convex-concave order, thereby generalising the
classical Strassen theorem. Another equivalent condition is that the
dual problem associated with the second-order Beckmann problem attains
its optimum at a \(\mathcal{C}^{1,1}(\mathbb{R}^n)\) function with
isometric derivative.
I will prove that the problem for \(\mu,\nu\) completely decomposes into a
collection of simpler problems on the leaves of the \(1\)-Lipschitz
derivative \(Du\) of an optimal solution
\(u\in\mathcal{C}^{1,1}(\mathbb{R}^n)\) for the dual problem. On each
such, leaf the solution is expressed in terms of bimartingale couplings
between conditional measures of \(\mu,\nu\), where the conditioning is
defined relative to the foliation induced by \(Du\).
Bibliography
\([1]\) Bołbotowski, K., Bouchitté, G. (2024) Kantorovich-Rubinstein duality theory for the Hessian, arXiv 2412.00516.