报告人：蒋继发（上海师范大学）

邀请人：吴付科

报告时间：2023年10月31日(星期二) 14:30-17:30

                 2023年11月02日(星期四) 14:30-17:30

                 2023年11月04日(星期六)  8:30-11:30

报告地点：科技楼南楼715

报告题目：On the Stochastic Stability of Invariant Measures

报告摘要：The lectures focus on the stochastic stability of invariant measures, which is closely related to the asymptotic measure problem proposed by Kolmogorov in the 1950s.

Lecture 1. Stochastic Stability of Invariant Measures for Quasipotential Systems

A quasipotential system is a vector field decomposed into the sum of a gradient vector field and a divergence-free vector field. Quasipotential systems are dense in smooth vector fields. This lecture gives criteria for a system to be quasipotential and for invariant measures to be stochastically stable under additive noise perturbation by the Laplace method. It is proved that limiting measures of stationary measures concentrate on the global minima set of the potential function. The Laplace method helps us to determine the weights on components of the global minima set, which gives the stochastically stable invariant measures. The supports of stochastically stable invariant measures contain either Lyapunov stable equilibria, or saddles, or periodic orbits, or quasiperiodic orbits or heteroclinic orbits, or even chaotic motions, while the existing results on gradient systems only contain Lyapunov stable equilibria.

Lecture 2. Uniform Large Deviation Principles with Applications to Stochastic Stability

This lecture presents a new criterion on uniform large deviation principle (ULDP) under rather mild conditions on the coefficients of the system by combining the weak convergent method and the Lyapunov method, which is much weaker than Freidlin-Wentzell's well-known classical criterion. Under mild Lyapunov conditions guaranteeing the existence of stationary measure and ULDP, we exploit limiting measures of stationary measures of stochastic ordinary differential equations. We prove that limiting measures are concentrated away from repellers and acyclic saddle or trap chains. This means limiting measures concentrate on Lyapunov stable invariant compact sets. Applications are made to the Morse-Smale systems and the Axiom A systems to get that stochastically stable sets are contained in Liapunov stable basic sets. The concept of the quasipotential will be introduced, such a version satisfies the Hamiltonian-Jacobian equation and is the leading order term of asymptotic expansion for the solution of Fokker-Planck equation. We will show how to compute them between limit sets by examples and generalize Freidlin-Wentzell's concentration result via our ULDP criterion. This is a joint work with Wang Jian, Zhai, Jianliang and Zhang, Tusheng.

Lecture 3. On the Stochastic Stability of the Liénard Equation

It is well-known that the Liénard equation can admit a unique limit cycle and multiple limit cycles. By construct suitable Lyapunov functions, we can prove the Liénard equation perturbed by suitable unbounded noise admits stationary measure and ULDP. We shall prove that the stable limit cycles are stochastically stable. If the stochastic equation is time-periodic, then it will admit stationary periodic process.

报告人简介：蒋继发，上海师范大学数理学院教授，博士生导师，上海市高峰项目《偏微分方程理论》学科方向负责人。1989年7月于中国科学院数学研究所获博士学位，获首届中国科学院院长奖学金特别奖。1992年被国家人事部评为有突出贡献的中青年专家，并获国务院政府特殊津贴；1993年分获安徽省科技进步奖二等奖和曾宪梓教育基金奖；1994年获中国科学技术大学首届《跨世纪优秀人才奖》；2004-2006连续三年被中国科学院评为“优秀研究生指导导师”；培养了两名博士生分获2004、2006年教育部全国百篇优秀论文。曾在中国科学技术大学任教十年，主持建立了全国第一个生物数学博士点。蒋继发教授最先把单调动力系统的研究引入我国，并一直是我国这一方向的学术带头人。曾主持多项国家自然科学基金委面上项目、参加多项国家自然科学基金委重点项目，主持多项国家教委（教育部）、科学院、上海市科委和教委项目。已发表130余篇SCI论文。