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.