Abstract: We study the continuum limits of Gibbs measures for periodic Toda lattices associated with its infinite many energies.    

We prove that, under suitable scaling, the Hamiltonian Gibbs measures of a class of FPU particle systems converge to independent white noises in $H^{-1/2-}(\mathbb T)^2$. For the Toda lattice and every fixed $m\ge1$, we construct conserved quantities converging to the corresponding Hamiltonians at level $m$ of the KdV hierarchy and prove convergence of their cutoff Gibbs measures in $H^{m-1/2-}(\mathbb T)^2$.

The main difficulty is to find the Toda conserved quantities that match the KdV hierarchy. To establish this correspondence, we derive a discrete Lenard recursion and develop an algebraic framework relating lattice differences to continuum derivatives.

This talk is based on a joint work with Guopeng Li (BIT), Tadahiro Oh (University of Edinburgh/BIT) and Yuanhong Tian (BIT).

Bio：郝子墨现任北京理工大学准聘教授，主要研究奇异漂移或奇异相互作用下的 McKean--Vlasov 方程、动理学(kinetic)方程，以及噪声正则化和随机系统的数值逼近问题。他于武汉大学获得数学学士学位，并分别在德国比勒费尔德大学和武汉大学完成博士培养，研究方向聚焦于奇异 McKean--Vlasov 方程与奇异动理学方程。2023年至2025年，他在比勒费尔德大学从事博士后研究，2026年起任北京理工大学准聘教授。郝子墨已在 The Annals of Probability、Communications in Mathematical Physics、Journal de Mathématiques Pures et Appliquées等国际期刊发表多篇论文。