中科院数学与系统科学研究院
数学研究所
学术报告
偏微分方程研讨班
时 间：2022.10.28（星期五）
地 点：思源楼S425
n 下午15:00-16:00
报告人：XICHENG ZHANG（Beijing Institute of Technology）
题 目：Second order Mean-field SDEs with singular kernel
摘 要：In this work we establish the local and global well-posedness of weak and strong solutions for second order mean-field SDEs with singular interaction kernels such as Newtonian or Coulomb potential, Riesz potential. Moreover, we also show the smoothness and the short time and long time asymptotic estimate of the density. Our results reveal a phenomena that for nonlinear mean-field equation, the regularity of the initial density could balance the singularity of the kernel. In particular, our results provide microscopic probability explanation for many physical models including Vlasov-Poisson-Fokker-Planck system, 2d-stochastic vortex system, surface quasi-geostropic models, etc. (This is an ongoing work with Zimo Hao.)
n 下午16:00-17:00
报告人：RONGCHAN ZHU （Beijing Institute of Technology）
题 目：A stochastic analysis approach to lattice Yang--Mills
摘 要：We develop a new stochastic analysis approach to the lattice Yang--Mills model at strong coupling in any dimension d>1, with t' Hooft scaling βN for the inverse coupling strength. We study their Langevin dynamics, ergodicity, functional inequalities, large N limits, and mass gap.
Assuming $|\beta| < \frac{N-2}{32(d-1)N}$ for the structure group SO(N), or $|\beta| < \frac{1}{16(d-1)}$ for SU(N), we prove the following results.The invariant measure for the corresponding Langevin dynamic is unique on the entire lattice, and the dynamic is exponentially ergodic under a Wasserstein distance.The finite volume Yang--Mills measures converge to this unique invariant measure in the infinite volume limit, for which Log-Sobolev and Poincar\'e inequalities hold.These functional inequalities imply that the suitably rescaled Wilson loops for the infinite volume measure has factorized correlations and converges in probability to deterministic limits in the large N limit, and correlations of a large class of observables decay exponentially, namely the infinite volume measure has a strictly positive mass gap. Our method improves earlier results or simplifies the proofs, and provides some new perspectives to the study of lattice Yang--Mills model.
