Mean-field interacting particle systems and McKean-Vlasov equation

Abstract

McKean-Vlasov quation describes heat diffusion in porous media and is the limit of propagation of chaos of system of interacting particle systems of mean field with a great number N of particles, known in 80s. Carrillo-McCann-Villani(03) renewed the subject by regarding MV equation as the gradient flow of the entropy functional on the space of probability measures w.r.t. the Otto- Villani's differential calculus, and proved the strict convexity of the entropy and so the entropical exponential convergence of MV equation by assuming the convexity of the confinement potential and the interacting potential. Their convexity assumption excludes the non-convex models which may have phase transition at low temperature, such as Curie-Weiss.

In this talk I will present some recent generalizations covering the non-convex cases on this subject, by using the tools such as Poincare inequality, log-Sobolev inequality, gradient estimate for interacting particle system. This talk is based on a series of joint works with A.Guillin, W. Liu, C. Zhang.

Speaker

哈尔滨工业大学数学研究院讲席教授。主要研究大偏差理论及其在数学物理、统计中的应用；算子半群的唯一性，谱分析；概率分析不等式。吴教授曾独立解决Varadhan猜测，量子场基态扩散过程的唯一性猜想，以及合作解决Gross猜想。他引入并建立一致可积算子概念及理论，L1和L∞的唯一性概念及理论等。这些工作在国际概率论界有很大影响。