摘 要：主要介紹了建立奇異攝動的擴散系統(tǒng)的隨機平均化原理的直接平均方法和當前的進展，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前在奇異攝動的延遲和泛函系統(tǒng)的隨機平均化原理中的進展和困難.

關鍵詞：奇異攝動；擴散系統(tǒng)；隨機平均化原理；鞅問題；弱收斂

中圖分類號：O211文獻標志碼：A

參 考 文 獻

［1］ ?GALTIER M，WAINRIB G.Multiscale analysis of slow-fast neuronal learning models with noise［J］.J Math Neurosci，2012，2 （1）：1-64.

［2］BAO J，YIN G，YUAN C.Two-time-scale stochastic partial differential equations driven by α-stable noise：averaging principles［J］.Bernoulli，2017，23：645-669.

［3］FREIDLIN M I，WENTZELL A D.Random Perturbations of Dynamical Systems［M］.3rd Edition.Berlin：Springer，2012.

［4］KHASMINSKII R Z，YIN G.On transition densities of singularly perturbed diffusions with fast and slow components［J］.SIAM J Appl Math，1996，56：1794-1819.

［5］PAVLIOTIS G A，STUART A M.Multiscale Methods：Averaging and Homogenization［M］.Berlin：Springer，2008.

［6］SKOROKHOD A V，HOPPENSTEADT F C，SALEHI H D.Random Perturbation Methods with Applications in Science and Engineering［M］.New York：Springer，2002.

［7］WU F，TIAN T，RAWLINGS J B，YIN G.Approximate method for stochastic chemical kinetics with two-time scales by chemical Langevin equations［J］.J Chemical Phy，2016，144（17）：174112.

［8］YIN G，ZHANG H Q.Singularly perturbed Markov chains：Limit results and applications［J］.Ann Appl Probab，2007，17：207-229.

［9］YIN G，ZHANG Q.Continuous-time Markov Chains and Applications：A Two-time-scale Approach［M］.New York：Springer，2013.

［10］KHASMINSKII R Z.On stochastic processes defined by differential equations with a small parameter［J］.Theory Probab Appl，1966，11：211-228.

［11］KHASMINSKII R Z，YIN G.Asymptotic series for singularly perturbed Kolmogorov-Fokker-Planck Equations［J］.SIAM J Appl Math，1996，56：1766-1793.

［12］KHASMINSKII R Z，YIN G.Limit behavior of two-time-scale diffusions revisited［J］.J Differential Equations，2005，212：85-113.

［13］KHASMINSKII R Z.Stochastic Stability of Differential Equations［M］.2nd Edition.Berlin：Springer，2012.

［14］PARDOUX E，VERETENNIKOV YU A.On the Poisson equation and diffusion approximation I［J］.Ann Probab，2001，29（3）：1061-1085.

［15］RCKNER M，XIE L.Diffusion approximation for fully coupled stochastic differential equations［J］.Ann Probab，2021，49（3）：1205-1236.

［16］KUSHNER H J.Approximation and Weak Convergence Methods for Random Processes，with Applications to Stochastic Systems Theory［M］.Cambridge：MIT Press，1984.

［17］KUSHNER H J. Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems［M］.Boston：Birkhuser，1990.

［18］KURTZ T.Semigroups of conditioned shifts and approximation of Markov processes［J］.Ann Probab，1975，3（4）：618-642.

［19］KARATZAS I，Shreve S E.Brownian Motion and Stochastic Calculus［M］.New York：Springer，1988.

［20］ZEIDLER E.Applied functional analysis：applications to mathematical physics［M］.New York：Springer Science & Business Media，2012.

［21］BILLINGSLEY P.Convergence of Probability Measures［M］.2nd Edition.New York：A Wiley-Interscience Publication，1999.

［22］MAO X.Stochastic Differential Equations and Applications［M］.2nd Edition.Chichester：［s.n.］，2007.

［23］BAO J，YIN G，YUAN C.Asymptotic analysis for functional stochastic differential equations［M］.London：Springer，2016.

［24］WU F，YIN G.Fast-slow-coupled stochastic functional differential［J］.J Differential Equations，2022，323：1-37.

［25］MOHAMMED S-E A.Stochastic Functional Differential Equations［M］.New York：［s.n.］，1986.

［26］WU F，YIN G.An averaging principle for two-time-scale stochastic functional differential equations［J］.J Differential Equations，2020，269：1037-1077.

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.

基金項目：國家自然科學基金（62273158）.

作者簡介（通信作者）：

吳付科（1976-），男，河南鄧州人，華中科技大學教授，博士，國家優(yōu)青，研究方向為隨機微分方程及其應用，E-mail：wufuke@hust.edu.cn.