学术报告—Two approaches to average stochastic perturbations of integrable systems

报 告 人：Sergei Kuksin，巴黎西岱大学教授

主 持 人：陈锋

时 间：2025年5月29日10：00

地 点：第六教学楼 911室

主办单位：长春大学数学与统计学院

报告人简介：Sergei Kuksin 教授 现任俄罗斯斯捷克洛夫数学研究所首席科学家、俄罗斯人民友谊大学数学实验室主任、法国巴黎西岱大学与索邦大学高级研究员。他的研究涵盖偏微分方程中的KAM理论、随机扰动偏微分方程、湍流与统计流体力学，以及紧致流形间函数的椭圆型偏微分方程。1992年他作为全会报告人出席巴黎欧洲数学家大会（ECM），1998年获邀在柏林国际数学家大会（ICM）作特邀报告，并荣获俄罗斯科学院颁发的李雅普诺夫奖。

观点综述：I will discuss

small stochastic perturbations of an integrable Hamiltonian ε -small stochastic perturbations of an integrable Hamiltonian system in R²ⁿ . Firstly I will write the perturbed equation using the action-angle variables of the integrable system, and formally average the obtained fast-slow system. The averaged equation for actions which we get in this way indeed describes the dynamics of the original equation for t ≤ Cε −1, where C is a constant, but only under some serious restrictions, which I will explain. A better way to study the long time dynamics of actions is inspired by the Krylov-Bogolyubov averaging: motivated by the latter, we guess in 𝑅 ²ⁿ a regular auxiliary equation, obtained by some averaging of the original one. Then we prove that under much weaker restrictions the actions of its solutions approximate those for solutions of the original equation for t ≤ Cε −1. Moreover, imposing some more restrictions on the equation we prove that this approximation holds uniformly in time.The talk is based on joint works with Andrey Piatnitski, Huang Guan and Guo Jing.