奇異攝動(dòng)的擴(kuò)散系統(tǒng)弱收斂意義下的平均化原理：直接平均法的概述與進(jìn)展

摘 要：主要介紹了建立奇異攝動(dòng)的擴(kuò)散系統(tǒng)的隨機(jī)平均化原理的直接平均方法和當(dāng)前的進(jìn)展，這種方法主要基于鞅問題和弱收斂.最后一部分也介紹了這種方法當(dāng)前在奇異攝動(dòng)的延遲和泛函系統(tǒng)的隨機(jī)平均化原理中的進(jìn)展和困難.

關(guān)鍵詞：奇異攝動(dòng)；擴(kuò)散系統(tǒng)；隨機(jī)平均化原理；鞅問題；弱收斂

中圖分類號(hào)：O211文獻(xiàn)標(biāo)志碼：A

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The averaged principle of diffusion systems with singular perturbations in the sense of weak convergence： overview and advancement of the direct-averaging method

Wu Fuke

（School of Mathematics and Statistics， Huazhong University of Science and Technology， Wuhan 430074， China）

Abstract： This paper mainly introduces the direct-averaging method for stochastic averaged principle of diffusion systems with singular perturbations， which is based on the martingale problem and the weak convergence. Finally， the advances and difficulties of this method in stochastic averaged principle of the diffusion delay and functional diffusion systems with singular perturbations.

Keywords： singular perturbation; diffusion system; stochastic averaged principle; martingale problem; weak convergence

［責(zé)任編校 陳留院 趙曉華］

收稿日期：2023-02-09;修回日期：2023-02-15.

基金項(xiàng)目：國(guó)家自然科學(xué)基金（62273158）.

作者簡(jiǎn)介（通信作者）：

吳付科（1976-），男，河南鄧州人，華中科技大學(xué)教授，博士，國(guó)家優(yōu)青，研究方向?yàn)殡S機(jī)微分方程及其應(yīng)用，E-mail：wufuke@hust.edu.cn.