随机平均原理研究若干进展*

许勇 裴斌 徐伟

(西北工业大学应用数学系,西安 710072)

随机平均原理研究若干进展*

许勇†裴斌 徐伟

(西北工业大学应用数学系,西安 710072)

本文介绍了随机平均原理的研究现状和发展趋势,探讨了基于非高斯列维噪声、分数高斯噪声、Markov切换的随机复杂动力学系统随机平均原理研究中的若干问题及进展.

非高斯列维噪声, 分数高斯噪声, Markov切换, 随机复杂动力学系统, 随机平均原理

引言

随机平均法以随机平均原理为理论基础是非线性随机动力学响应分析的重要工具,分为标准随机平均法和能量包线随机平均法两种,其中标准随机平均法主要应用在多自由度拟线性随机系统中,而能量包线随机平均法主要应用在多自由度强非线性拟保守随机系统中.随机平均法凭借其简单、可以降维、效率高等优点在动力学研究中被广泛应用.因此,对于随机平均原理的研究也就成为一项具有重要科学意义和实际指导价值的研究.

在实际应用中,很多系统的动力学模型是既包含动力学时间尺度较快的状态变量,又包含时间尺度较慢的状态变量的.若引入适当的无量纲参数来表示不同动力学时间尺度的比值,则这些系统可表示成由快变量和慢变量相耦合的系统,即快-慢(两尺度)动力系统.快-慢系统中复杂的动力学现象得到广泛关注.例如,在许多工程技术领域中的控制问题；在生态系统中,生态环境的恶化、物种的爆发和消亡所产生的动力学机制已得到深入研究；在生物神经系统中,存在各种快-慢过程,使得系统存在各种形式的分岔和丰富的放电模式.在实际科学中,许多问题可以转换成研究系统的两个时间尺度,如出现在应用程序中不同的化学反应动力学[1,2],细胞建模[3,4],哈密顿系统[5,9],电子电路[10,11]和激光系统[12,14]. 最著名的快-慢系统可以追溯到范德波[10]在1920年提出的范德波方程. 基于平均原理的平均法是分析快-慢系统动力学行为的有效工具, 其目的在于构造一个所谓的“平均化方程” (也称为“简化方程”或“有效方程”) 来简化原来的多尺度方程, 使简化后的方程不再包含快尺度物理量, 并且使得简化方程的解可以逼近原来方程中慢尺度的物理量.具体来看,考虑一个具有快-慢两个尺度的常微分方程：

(1)

假设对任意x∈Rn,极限：

(2)

的解轨道一致逼近(当参数ε趋于0时).确定性方程平均化原理的研究有较长的历史,其奠基性工作由前苏联数学家Bogoliubov 在文献[15]中完成.紧接着,Gikhman[16], Volosov[17]和Besjes[18]研究了非线性常微分方程的平均化问题. 随机平均原理首先由Stratonovich[19]提出,此后,Khasminskii[20,22]将平均化原理发展到具有快-慢时间尺度的随机常微分方程的研究中,他在文献[23]中证明了随机平均化原理在较弱的收敛意义下成立.值得一提的是,Veretenniko[24,25],Freidlin & Wentzell[26,27]进一步显著改进了Khasminskii的结果,将较弱收敛意义下的随机平均化原理推广到依概率收敛的情形.另外,文献Golec & Ladde[28], Givon,Kevrekidis& Kupferman[29]研究了关于均方收敛意义下的随机平均化原理,文献Golec[30]和Givon[29]得到了强收敛意义下的随机平均化原理.Zhu[31]和Roberts和Spanos[32]等人的专著及综述中都对随机平均法早期的发展做了详细介绍. Zhu[5,9]的团队提出并发展了高斯白噪声、谐和噪声等作用下单自由度或多自由度拟Hamilton系统的随机平均理论和方法,解决了五类拟Hamilton系统平均方程的求解问题.Xu[33]等建立了高斯色噪声驱动下一类随机动力学系统的平均原理及高斯白噪声与色噪声共同激励下一类单自由度系统的随机平均法[34].

近几年来,随机平均法理论得到进一步完善,并已被应用于各类随机动力系统动力学性质的研究,其为研究更复杂的随机动力学系统和解决各种激励下的随机动力学问题提供了良好的方法.随机平均的方法和理论也不再仅仅基于以高斯白噪声为代表的不相关噪声激励下的随机动力学系统,具有相关时间的分数高斯噪声激励、非高斯列维噪声及Markov切换的随机动力学系统(包含无穷维系统)研究越来越引起学者的关注,取得了一定的发展.

本文根据国内外研究现状和发展趋势,综述了基于非高斯列维噪声、长相关性分数高斯噪声及Markov切换的随机复杂动力学系统(包含无穷维系统)随机平均原理研究中的若干研究方向,并对存在的一些问题以及进一步的研究做了展望.

1 基于非高斯列维噪声的随机平均原理

在以往的大部分研究中,为了处理起来简便,研究人员考虑的都是高斯噪声,它是布朗运动的形式导数,一般用来描述连续型的微小的随机因素.在大多数情况下,高斯的假设是比较合理的,它满足中心极限定理,而且由于处理起来比较简单,理论推导比较容易,在许多领域都得到了广泛的应用.然而,高斯噪声只是一种理想的噪声源,它刻画的是正常扩散,即只能模拟均值在小范围内的起伏,而不能模拟大幅度的涨落.在实际应用中,我们遇到的许多噪声都是非高斯的,比如在生物医学中的诱发电位噪声、低频的大气噪声以及各种其它人为噪声等.这些噪声的非高斯性使得它们具有更强的冲击性,其所服从的分布比起正态分布,具有更多的尖峰与偶然性 (见图1),而且其密度函数的拖尾与高斯密度函数相比,衰减的也更为缓慢 (见图2)[35].这种情况下,以往基于高斯假定所得到的结论就需要被重新考虑,我们需要寻求一种更加广义,能够更好的与实际符合的分布,它的导数能更好地用来描述我们所遇到的噪声.

图1 不同的稳定性指标对应的列维噪声的概率密度函数Lα,β(ζ;D,μ)Fig.1 Probability density functions Lα,β(ζ;D,μ) for Lévy noise with different stability indexes

图2 不同的偏斜参数对应的列维噪声的概率密度函数Lα,β(ζ;D,μ),α=1.2Fig.2 Probability density functions Lα,β(ζ;D,μ),α=1.2 for Lévy noise with different skewness parameters

Zhu[36]首先将随机平均法运用到泊松白噪声激励下的非线性系统的研究中,Zeng 和Zhu[37,40]研究了非高斯随机激励下非线性系统的随机平均法.Xu[41]给出了非高斯列维噪声驱动下的随机动力系统的平均原理,Xu[42]还给出了在一类弱化的李普希兹条件下非高斯列维噪声驱动下的随机动力系统的平均原理,证明了平均后随机动力学系统的解依概率和均方收敛于原系统的解,给出了随机平均法的理论依据. Givon[43]根据快变量存在的不变测度,研究了两尺度跳扩散过程均方意义下的随机平均原理,并得到相应的收敛阶为O(lnε)：

(3)

(4)

2 基于分数高斯噪声的随机平均原理

在自然界等很多现象中的噪声往往表现出相关性甚至是长相关性的显著特征,而分数布朗运动为长相关性噪声的研究提供了重要的理论基础,它是一种比布朗运动更广泛的随机过程,具有的长相关性、增量非独立性已经在金融[48,49]、地球物理学[50,51]、生物学[52,53]和脑功能信号分析[54,55]等方面有了一定的应用.1968年,Mandelbrot和Van Ness[56]首先定义了“分数布朗运动”,并给出分数布朗运动的构造.此后,分数布朗运动驱动随机动力系统的研究引起学者的关注.由于分数布朗运动既不是半鞅又不是马尔可夫过程,使得随机积分这个完备的理论基础并不适用于分数布朗运动的研究.Xu[57-59]在前向路径积分意义下,根据Khasminskii平均法,研究具有长相关性分数布朗运动的随机平均原理,证明了具有长相关性分数布朗运动驱动的动力系统与平均后的随机动力系统在均方意义下是收敛的,并利用数值模拟的方法,验证了定理的正确性.Xu[60,61]还进一步研究了分数布朗运动驱动的快-慢变系统的随机平均原理.Deng和Zhu[62,64]根据分数布朗运动驱动的两尺度随机动力系统随机平均原理结果[60,61],提出并发展了分数高斯噪声等作用下单自由度或多自由度拟Hamilton系统的随机平均理论和方法,解决了拟Hamilton系统平均方程的求解问题.

3 基于Markov切换的随机平均原理

1961年Krasovskii和Lidskii[65]首次提出Markov切换系统.近些年来,Markov切换系统在复杂网络等非线性系统建模中起不可替代的作用,使得其迅速成为国内外发展最活跃的前沿学科和研究热点之一.所谓Markov切换系统,即是以一个连续时间有限状态的Markov链来控制系统的切换时刻和切换状态.由于Markov切换系统贴近应用背景,数学描述清晰,可操作性强,因而很快成为复杂网络及其他非线性系统建模的重要参考依据.事实上,自然界和人类社会中广泛存在着的各种各样的复杂系统都可以通过复杂网络模型来描述.复杂网络模型描述了复杂系统中元素之间、子系统之间、层次之间的相互作用以及系统与环境的相互作用.而当系统元件出现突发故障或突然修复状况,或突然出现外部干扰,以及子系统连接方式发生突变时,复杂网络系统很可能产生结构或参数上的突发改变.例如种群系统中的环境噪音和地震等突发性现象对种群数量变化的影响；通讯系统中数据交换的障碍和机器故障对数据传输的影响；经济金融系统中,国家宏观调控对经济变化的影响等等.考虑到Markov切换系统在描述系统结构或参数突然变化方面具有很大的优势,因此,选用Markov切换系统来描述网络的状态更加贴合实际应用背景.具有Markov切换的随机动力系统的随机平均原理得到了广泛的研究.Yin[66]研究了具有两尺度Markov切换的跳扩散模型的随机平均原理:

dXε(t)=f(Xε(t),αε(t),t)dt+

g(Xε(t),αε(t),t)dw(t)+

∫Γh(Xε(t),αε(t),t,γ)N(dt,dγ)

(5)

其中αε(t)代表连续时间两尺度Markov切换.以及快变量慢变量耦合的Markov切换调制两尺度随机跳扩散过程的随机平均原理:

(6)

(7)

其中αε(t)代表连续时间两尺度Markov切换,v(t)表示Wiener过程. Bao,Yin和Yuan[68]考虑了加性α稳定噪声激励的两尺度随机微分方程的随机平均原理.宦荣华教授和朱位秋院士[69-70]等人给出了具有Markov切换的随机平均法,并用此方法研究了Markov切换多自由度随机拟不可积哈密顿系统的概率1稳定性和在时滞反馈控制下Markov切换拟可积哈密顿系统的概率1稳定性.

4 无穷维随机动力系统的随机平均原理研究

5 结语

本文仅就我们关心的领域对非高斯列维噪声、分数高斯噪声、Markov切换,及无穷维随机系统随机平均原理的研究做了介绍,还有许多方面没有涉及.

目前来说,对于高斯白噪声、列维噪声激励的有穷维系统的随机平均原理较为系统,已经存在大量成熟的结果,而对于列维噪声、分数高斯噪声激励的无穷维系统及含Markov切换的随机系统的随机平均原理的研究尚在起步阶段,研究还相当地少,特别是针对于乘性列维噪声、分数高斯噪声激励的无穷维系统及含两尺度Markov切换的随机平均原理的研究是未来的重点.

1Koper M T M. Bifurcations of mixed-mode oscillations in a three variable autonomous Van der Pol-Duffing model with a cross-shaped phase diagram.PhysicaD, 1995,80(1):72～94

2Larter R, Steinmetz C G, Aguda B D. Fast-slow variable analysis of the transition to mixed-mode oscillations and chaos in the peroxidase reaction.JournalofChemicalPhysics, 1988,89(10):6506～6514

3Krupa M, Popovic N, Kopell N and Rotstein H G. Mixed-mode oscillations in a three time-scale model for the dopaminergic neuron.Chaos, 2008,18(1):015106

4Rubin J, Wechselberger M. Giant squid-hidden canard: the 3D geometry of the Hodgkin-Huxley model.BiologicalCybernetics, 2007,97(1):5～32

5朱位秋. 非线性随机动力学与控制-Hamilton理论体系框架. 北京:科学出版社, 2003 (Zhu W Q. Nonlinear stochastic dynamics and control-Hamiltonian theoretical framework. Beijing: Science Press, 2003 (in Chinese))

6Zhu W Q, Huang Z L, Suzuki Y. Stochastic averaging and Lyapunov exponent of quasi partially integrable Hamiltonian systems.InternationalJournalofNon-LinearMechanics, 2002,37(3):419～437

7Zhu W Q, Yang Y Q. Stochastic averaging of quasi-nonintegrable Hamiltonian systems.ASMEJournalofAppliedMechanics, 1997,64(1):157～164

8Zhu W Q, Huang Z L, Yang Y Q. Stochastic averaging of quasi-integrable Hamiltonian systems.ASMEJournalofAppliedMechanics, 1997,64(4):975～984

9Huang Z L, Zhu W Q. Stochastic averaging of quasi-generalized Hamiltonian systems.InternationalJournalofNon-LinearMechanics, 2009,44(1):71～80

10 Van der Pol B. A theory of the amplitude of free and forced triode vibrations.RadioReview, 1920,1:701～710

11 Bernd K, Hinke M O, Jorge G V. Numerical continuation methods for dynamical systems: Path following and boundary value problems, Dordrecht: Springer，2007:221～251

12 Braza P, Erneux E. Singular Hopf bifurcation to unstable periodic solutions in an NMR laser.PhysicalReviewA, 1989,40(5):2539

13 Dubbeldam J L, Krauskopf B. Self-pulsations in lasers with saturable absorber: dynamics and bifurcations. Optics Communications, 1999,159(4-6):325～38

14 Erneux T, Mandel P. Bifurcation phenomena in a laser with a saturable absorber.ZeitschriftfürPhysikBCondensedMatter, 1981,44(4):365～374

15 Bogoliubov N N, Mitropolsky Y A. Asymptotic Methods in the Theory of Nonlinear Oscillations. New York: Gordon & Breach Science Publishers, 1961

16 Sorokina N G. On a theorem of N N Bogoliubov. Ukrain. Math. Zhurn, 1959,11:220～222

17 Volosov V M. Averaging in systems of ordinary differential equations.RussianMathematicalSurveys, 1962,17(6):1～126

18 Besjes J G. On the asymptotic methods for non-linear differential equations.JournalDeMecanique, 1969,8(3):357～373

19 Stratonovich R L. Topics in the theory of random noise, New York, Gordon and Breach, 1963,1967,2

20 Khasminskii R. On stochastic processes defined by differential equations with a small parameter.TheoryofProbability&ItsApplications, 1966,11(2):211～228

21 Khasminskii R. A limit theorem for the solutions of differential equations with random right-hand sides.TheoryofProbability&ItsApplications, 1966,11(3):390～406

22 Khasminskii R Z. Principle of averaging for parabolic and elliptic differential equations and for Markov processes with small diffusion.TheoryofProbability&ItsApplications, 1963,8(1):1～21

23 Khasminskii R Z. On the principle of averaging the Ito stochastic differential equations.Kibernetika, 1968,4:260～279

24 Veretennikov A Y. On the averaging principle for systems of stochastic differential equations.MathematicsoftheUSSR-Sbornik, 1991,69(1):271～284

25 Veretennikov A. Y. On large deviations in the averaging principle for SDEs with full dependence.AnnalsofProbability, 1999,27(1):284～296

26 Freidlin M I, Wentzell A D. Long-time behavior of weakly coupled oscillators.JournalofStatisticalPhysics, 2006,123(6):1311～1337

27 Freidlin M I, Wentzell A D. Random perturbations of dynamical systems, Springer, 1998

28 Golec J, Ladde G. Averaging principle and systems of singularly perturbed stochastic differential equations.JournalofMathematicalPhysics, 1990,31(5):1116～1123

29 Givon D, Kevrekidis I G, Kupferman R. Strong convergence of projective integration schemes for singular perturbed stochastic differential systems.CommunicationsinMathematicalSciences, 2006,4(4):707～729

30 Golec J. Stochastic averaging principle for systems with pathwise uniqueness.StochasticAnalysis&Applications, 1995,13(3):307～322

31 Zhu W Q. Stochastic averaging methods in random vibration.ASMEAppliedMechanicsReviews, 1988,41(5):189～199

32 Roberts J B, Sranos P D. Stochastic averaging: an approximate method of solving random vibration problems.InternationalJournalofNon-LinearMechanics, 1986,21(2):111～134

33 Xu Y, Gu R C, Zhang H Q, Xu W, Duan J Q. Stochastic bifurcations in a bistable Duffing-Van der Pol oscillator with colored noise.PhysicalReviewE, 2011,83(5):056215

34 Xu Y, Guo R, Jia W T, Li J J. Stochastic averaging for a class of single degree of freedom systems with combined Gaussian noises.ActaMechanica, 2014,225(9):2611～2620

35 Xu Y, Li Y, Zhang H, et al. The Switch in a genetic toggle system with Lévy noise.ScientificReports, 2016,6:31505

36 Zeng Y, Zhu W Q. Stochastic averaging of quasi-non-integrable Hamiltonian systems under Poisson white noise excitation.ASMEJournalofAppliedMechanics, 2011,78(2):856～875

37 曾岩. 非高斯随机激励下非线性系统的随机平均法[博士学位论文]. 杭州:浙江大学, 2010 (Zeng Y. Stochastic averaging method of nonlinear system under non-gaussian random excitations[PhD Thesis]. Hangzhou: Zhejiang University, 2010 (in Chinese))

38 Zeng Y, Zhu W Q. Stochastic averaging of n-dimensional non-linear dynamical systems subject to non-Gaussian wide-band random excitations.InternationalJournalofNon-LinearMechanics, 2010,45(5):572～586

39 Zeng Y, Zhu W Q. Stochastic averaging of quasi-linear systems driven by Poisson white noise.ProbabilisticEngineeringMechanics, 2010,25(1):99～107

40 Zeng Y, Zhu W Q. Stochastic averaging of strongly nonlinear oscillators under poisson white noise excitation.IUTAMSymposiumonNonlinearStochasticDynamicsandControl, 2011,29(2):147～155

41 Xu Y, Duan J, Xu W. An averaging principle for stochastic dynamical systems with Lévy noise.PhysicaD, 2011,240(17):1395～1401

42 Xu Y, Pei B, Li Y G. Approximation properties for solutions to non-Lipschitz stochastic differential equations with Lévy noise.MathematicalMethodsintheAppliedSciences, 2015,38 (11):2120～2131

43 Givon D. Strong convergence rate for two-time-scale jump-diffusion stochastic differential systems.SiamJournalonMultiscaleModeling&Simulation, 2007,6(2):577～594

44 Liu D. Strong convergence of principle of averaging for multiscale stochastic dynamical systems.CommunicationsinMathematicalSciences. 2010,8(4):999～1020

45 Liu D. Strong convergence rate of principle of averaging for jump-diffusion processes.FrontiersofMathematicsinChina, 2012,7(2):305～320

46 Xu J, Miao Y.Lp(p>2)-strong convergence of an averaging principle for two-time-scales jump-diffusion stochastic differential equations.NonlinearAnalysis:HybridSystems, 2015,18:33～47

47 Thompson W, Kuske R, Monahan A. Stochastic averaging of dynamical systems with multiple time scales forced with α-stable noise.SiamJournalonMultiscaleModeling&Simulation, 2015,13(4):1194～1223

48 Backus D K, Zin S E. Long-memory inflation uncertainty: evidence from the term structure of interest rate.JournalofMoneyCreditandBanking, 1993,25(3):701～708

49 Rostek S. Option Pricing in Fractional Brownian Markets, Springer, Berlin, 2009

50 Granger C W, Joyeux R. An Introduction to long-memory time series models and fractional differencing.JournalofTimeSeriesAnalysis, 2010,1(1):15～29

51 Mandelbrot B B, Wallis J R. Some long-run properties of geophysical records.WaterResourcesResearch, 1969,5(2):321～340

52 Peng C K, Havlin S V, et al. Mosaic organization of DNA nucleotides.PhysicalReviewE, 1994,49(2):1685

53 Roche S, Bicout D, et al. Long range correlations in DNA: scaling properties and charge transfer efficiency.PhysicalReviewLetters, 2003,91(22):228101

54 Maxim V, Sendur L, et al. Fractional Gaussian noise functional MRI and Alzheimer′s disease.Neuroimage, 2005,25(1):141～58.

55 Wink A M, Bernard F, et al. Age and cholinergic effects on hemodynamics and functional coherence of human hippocampus.NeurobiologyofAging, 2006,27(10):1395

56 Mandelbrot B B, Van Ness J W. Fractional brownian motions, fractional noises and applications.SiamReview, 1968,10(4):422～437

57 Xu Y, Guo R, et al. Stochastic averaging principle for dynamical systems with fractional Brownian motion.DiscreteandContinuousDynamicalSystems-SeriesB, 2014,19(4):1197～1212

58 Xu Y, Pei B, Wu J L. Stochastic averaging principle for differential equations with non-Lipschitz coefficients driven by fractional Brownian motion.StochasticsandDynamics, 2017,17:1750013

59 Xu Y, Pei B, Li Y G. An averaging principle for stochastic differential delay equations with fractional Brownian motion.AbstractandAppliedAnalysis, 2013,2014(1):1～10

60 Xu Y, Pei B, Guo R. Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion.DiscreteandContinuousDynamicalSystemsSeriesB, 2015,20(7):2257～2267

61 Xu Y, Guo R, et al. A limit theorem for the solutions of slow-fast systems with fractional Brownian motion.TheoreticalandAppliedMechanicsLetters, 2014,4(1):22～25

62 Lü Q F, Deng M L, Zhu W Q. Stochastic averaging of quasi integrable and resonant Hamiltonian systems excited by fractional Gaussian noise with Hurst index 1/2

63 Deng M L, Zhu W Q. Responses of linear and nonlinear oscillator to fractional Gaussian noise with Hurst index between 1/2 and 1.JournalofAppliedMechanics, 2015,82(10):101008

64 Deng M L, Zhu W Q. Stochastic averaging of quasi-non-integrable Hamiltonian systems under fractional Gaussian noise excitation.NonlinearDynamics., 2016,83(1-2):1015～1027

65 Kats I, Krasovskii N. On the stability of systems with random parameters.JournalofAppliedMathematicsandMechanics, 1960,24(5):1225～1246

66 Yin G, Yang H. Two-time-scale jump-diffusion models with Markovian switching regimes.StochasticsandStochasticReports, 2004,76(2):77～99

67 Yin G, Talafha Y, Xi F. Stochastic Liénard equations with random switching and two-time scales.CommunicationsinStatistics-TheoryandMethods, 2014,43(7):1533～1547

68 Bao J, Yin G, Yuan C. Two-time-scale stochastic partial differential equations driven by alpha-stable noises: Averaging principles.Bernoulli, 2017,23(1):645～669

69 Huan R H, Zhu W, Ma F, et al. Stationary response of a class of nonlinear stochastic systems undergoing Markovian jumps.ASMEJournalofAppliedMechanics, 2015,82(5):051008

70 Huan R H, Zhu W Q, Ma F, et al. Asymptotic stability of a class of nonlinear stochastic systems undergoing Markovian jumps.ProbabilisticEngineeringMechanics, 2016,45(5):13～21

71 Cerrai S. A khasminskii type averaging principle for stochastic reaction-diffusion equations.AnnalsofAppliedProbability, 2008,19(3):899～948

72 Cerrai S, Freidlin M. Averaging principle for a class of stochastic reaction-diffusion equations.ProbabilityTheory&RelatedFields, 2009,144(1-2):137～177

73 Wang W, Roberts A J. Average and deviation for the stochastic FitzHugh-Nagumo system.Australian&NewZealandIndustrial&AppliedMathematicsJournal, 2008,50:292～307

74 Fu H, Duan J. An averaging principle for two time-scales stochastic partial differential equations.StochasticandDynamics, 2011,11:353～367.

75 Fu H, Liu J. Strong convergence in stochastic averaging principle for two time-scales stochastic partial differential equations.JournalofMathematicalAnalysis&Applications, 2011,384(1):70～86

76 Xu J, Miao Y, Liu J. Strong averaging principle for slow-fast SPDEs with Poisson random measures.DiscreteandContinuousDynamicalSystems:B, 2015,20(7):2233～2256

77 Xu J. Lp-strong convergence of the averaging principle for slow-fast SPDEs with jumps.JournalofMathematicalAnalysisandApplications, 2017,445(1):342～373

78 Pei B, Xu Y, Wu J L. Two-time-scales hyperbolic-parabolic equations driven by Poisson random measures: Existence, uniqueness and averaging principles.JournalofMathematicalAnalysisandApplications, 2017,447(1):243～268

*The project supported by the National Natural Science Foundation of China(11372247, 11572247)

† Corresponding author E-mail:hsux3@nwpu.edu.cn

3 April 2017,revised 18 April 2017.

SOME RECENT DEVELOPMENTS OF STOCHASTIC AVERAGING PRINCIPLE*

Xu Yong†Pei Bin Xu Wei

(DepartmentofAppliedMathematics,NorthwesternPolytechnicalUniversity,Xi′an710072,China)

Based on the recent research status and development trend of the stochastic averaging principle, we discussed the averaging principles with respect to non-Gaussian Lévy noise, fractional Gaussian noise and complex dynamics system with Markov switching.

non-Gaussian Lévy noise, fractional Gaussian noise, Markov switching, stochastic complex dynamics system, stochastic averaging principle

*国家自然科学基金资助项目(11372247, 11572247)

10.6052/1672-6553-2017-022

2017-04-03收到第1稿,2017-4-21收到修改稿.

† 通讯作者 E-mail:hsux3@nwpu.edu.cn