动力系统研讨班：On the Kolmogorov Asymptotic Measure Problem of Stochastic Ordinary Differential Equations

中科院数学与系统科学研究院

数学研究所

学术报告

动力系统研讨班

报告人： 蒋继发 教授 (上海师范大学)

题  目：On the Kolmogorov Asymptotic Measure Problem of Stochastic Ordinary Differential Equations

时  间：2024.01.10（星期三）16:00-17:00

地  点：数学院南楼N913

摘  要：This talk will focus on Kolmogorov's asymptotic measure problem of stationary measures for stochastic ordinary differential equations. For a quasipotential system with additive noise, we first prove that the  asymptotic measure of stationary measures is concentrated on the global minima set of its potential, under nondegenerate conditions on the Hessian matrices of the potential at global minimal points or components, the asymptotic measure is precisely obtained and its support could be stable equilibria, or saddles, or periodic orbits, or quasiperiodic orbits, or contains Smale horseshoe with uncountable chaotic orbits. Using uniform large deviations principle even with degenerate diffusion, we then present a method how to estimate rare probabilities of small neighborhoods of unstable invariant sets. This proves asymptotic measures preclude concentration on repellers and acyclic saddles and it must support on Lyapunov stable compact invariant sets. Freidlin-Wentzell's theorem on concentration of asymptotic measures is generalized to the case of unbounded coefficients. Besides, we prove that stationary measures of stochastic van der Pol equation with unbounded and degenerate noise converge weakly to the normalized arc-length measure of the unique limit cycle. The second part is a joint work with Wang Jian, Zhai Jianliang and Zhang Tusheng.