學術報告—Two approaches to average stochastic perturbations of integrable systems

報 告 人：Sergei Kuksin，巴黎西岱大學教授

主 持 人：陳鋒

時 間：2025年5月29日10：00

地 點：第六教學樓 911室

主辦單位：長春大學數學與統計學院

報告人簡介：Sergei Kuksin 教授 現任俄羅斯斯捷克洛夫數學研究所首席科學家、俄羅斯人民友誼大學數學實驗室主任、法國巴黎西岱大學與索邦大學高級研究員。他的研究涵蓋偏微分方程中的KAM理論、隨機擾動偏微分方程、湍流與統計流體力學，以及緊致流形間函數的橢圓型偏微分方程。1992年他作為全會報告人出席巴黎歐洲數學家大會（ECM），1998年獲邀在柏林國際數學家大會（ICM）作特邀報告，并榮獲俄羅斯科學院頒發的李雅普諾夫獎。

觀點綜述：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 ?? 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.