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.

The Brownian Marble

Andreas Kyprianou (University of Warwick)

Let \(R:(0,\infty) \to [0,\infty)\) be a measurable function. Consider a family of coalescing Brownian motions started from every point in the subset \(\{ (0,x) : x \in \mathbb{R} \}\) of \([0,\infty) \times \mathbb{R}\) and proceeding according to the following rule: the interval \(\{t\} \times [L_t,U_t]\) between two consecutive Brownian motions instantaneously ‘fragments’ at rate \(R(U_t - L_t)\). At such a fragmentation event at a time \(t\), we initiate new coalescing Brownian motions from each of the points \(\{ (t,x) : x \in [L_t,U_t]\}\). The resulting process, which we call the \(R\)-marble, is easily constructed when \(R\) is bounded, and may be considered a random subset of the Brownian web.

Under mild conditions, we show that it is possible to construct the \(R\)-marble when \(R\) is unbounded as a limit as \(n \to \infty\) of \(R_n\)-marbles where \(R_n(g) = R(g) \wedge n\). The behaviour of this limiting process is mainly determined by the shape of \(R\) near zero. The most interesting case occurs when the limit \(\lim_{g \downarrow 0} g^2 R(g) = \lambda\) exists in \((0,\infty)\), in which case we find a phase transition. For \(\lambda \geq 6\), the limiting object is indistinguishable from the Brownian web, whereas if \(\lambda < 6\), then the limiting object is a nontrivial stochastic process with large gaps.

When \(R(g) = \lambda/g^2\), the \(R\)-marble is a self-similar stochastic process which we refer to as the Brownian marble with parameter \(\lambda > 0\). We give an explicit description of the space-time correlations of the Brownian marble, which can be described in terms of an object we call the Brownian vein; a spatial version of a recurrent extension of a killed Bessel-\(3\) process.