隨機(jī)平均原理研究若干進(jìn)展*

許勇 裴斌 徐偉

(西北工業(yè)大學(xué)應(yīng)用數(shù)學(xué)系,西安 710072)

隨機(jī)平均原理研究若干進(jìn)展*

許勇?裴斌 徐偉

(西北工業(yè)大學(xué)應(yīng)用數(shù)學(xué)系,西安 710072)

本文介紹了隨機(jī)平均原理的研究現(xiàn)狀和發(fā)展趨勢(shì),探討了基于非高斯列維噪聲、分?jǐn)?shù)高斯噪聲、Markov切換的隨機(jī)復(fù)雜動(dòng)力學(xué)系統(tǒng)隨機(jī)平均原理研究中的若干問(wèn)題及進(jìn)展.

非高斯列維噪聲, 分?jǐn)?shù)高斯噪聲, Markov切換, 隨機(jī)復(fù)雜動(dòng)力學(xué)系統(tǒng), 隨機(jī)平均原理

引言

隨機(jī)平均法以隨機(jī)平均原理為理論基礎(chǔ)是非線性隨機(jī)動(dòng)力學(xué)響應(yīng)分析的重要工具,分為標(biāo)準(zhǔn)隨機(jī)平均法和能量包線隨機(jī)平均法兩種,其中標(biāo)準(zhǔn)隨機(jī)平均法主要應(yīng)用在多自由度擬線性隨機(jī)系統(tǒng)中,而能量包線隨機(jī)平均法主要應(yīng)用在多自由度強(qiáng)非線性擬保守隨機(jī)系統(tǒng)中.隨機(jī)平均法憑借其簡(jiǎn)單、可以降維、效率高等優(yōu)點(diǎn)在動(dòng)力學(xué)研究中被廣泛應(yīng)用.因此,對(duì)于隨機(jī)平均原理的研究也就成為一項(xiàng)具有重要科學(xué)意義和實(shí)際指導(dǎo)價(jià)值的研究.

在實(shí)際應(yīng)用中,很多系統(tǒng)的動(dòng)力學(xué)模型是既包含動(dòng)力學(xué)時(shí)間尺度較快的狀態(tài)變量,又包含時(shí)間尺度較慢的狀態(tài)變量的.若引入適當(dāng)?shù)臒o(wú)量綱參數(shù)來(lái)表示不同動(dòng)力學(xué)時(shí)間尺度的比值,則這些系統(tǒng)可表示成由快變量和慢變量相耦合的系統(tǒng),即快-慢(兩尺度)動(dòng)力系統(tǒng).快-慢系統(tǒng)中復(fù)雜的動(dòng)力學(xué)現(xiàn)象得到廣泛關(guān)注.例如,在許多工程技術(shù)領(lǐng)域中的控制問(wèn)題；在生態(tài)系統(tǒng)中,生態(tài)環(huán)境的惡化、物種的爆發(fā)和消亡所產(chǎn)生的動(dòng)力學(xué)機(jī)制已得到深入研究；在生物神經(jīng)系統(tǒng)中,存在各種快-慢過(guò)程,使得系統(tǒng)存在各種形式的分岔和豐富的放電模式.在實(shí)際科學(xué)中,許多問(wèn)題可以轉(zhuǎn)換成研究系統(tǒng)的兩個(gè)時(shí)間尺度,如出現(xiàn)在應(yīng)用程序中不同的化學(xué)反應(yīng)動(dòng)力學(xué)[1,2],細(xì)胞建模[3,4],哈密頓系統(tǒng)[5,9],電子電路[10,11]和激光系統(tǒng)[12,14]. 最著名的快-慢系統(tǒng)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-慢系統(tǒng)動(dòng)力學(xué)行為的有效工具, 其目的在于構(gòu)造一個(gè)所謂的“平均化方程” (也稱為“簡(jiǎn)化方程”或“有效方程”) 來(lái)簡(jiǎn)化原來(lái)的多尺度方程, 使簡(jiǎn)化后的方程不再包含快尺度物理量, 并且使得簡(jiǎn)化方程的解可以逼近原來(lái)方程中慢尺度的物理量.具體來(lái)看,考慮一個(gè)具有快-慢兩個(gè)尺度的常微分方程：

(1)

假設(shè)對(duì)任意x∈Rn,極限：

(2)

的解軌道一致逼近(當(dāng)參數(shù)ε趨于0時(shí)).確定性方程平均化原理的研究有較長(zhǎng)的歷史,其奠基性工作由前蘇聯(lián)數(shù)學(xué)家Bogoliubov 在文獻(xiàn)[15]中完成.緊接著,Gikhman[16], Volosov[17]和Besjes[18]研究了非線性常微分方程的平均化問(wèn)題. 隨機(jī)平均原理首先由Stratonovich[19]提出,此后,Khasminskii[20,22]將平均化原理發(fā)展到具有快-慢時(shí)間尺度的隨機(jī)常微分方程的研究中,他在文獻(xiàn)[23]中證明了隨機(jī)平均化原理在較弱的收斂意義下成立.值得一提的是,Veretenniko[24,25],Freidlin & Wentzell[26,27]進(jìn)一步顯著改進(jìn)了Khasminskii的結(jié)果,將較弱收斂意義下的隨機(jī)平均化原理推廣到依概率收斂的情形.另外,文獻(xiàn)Golec & Ladde[28], Givon,Kevrekidis& Kupferman[29]研究了關(guān)于均方收斂意義下的隨機(jī)平均化原理,文獻(xiàn)Golec[30]和Givon[29]得到了強(qiáng)收斂意義下的隨機(jī)平均化原理.Zhu[31]和Roberts和Spanos[32]等人的專著及綜述中都對(duì)隨機(jī)平均法早期的發(fā)展做了詳細(xì)介紹. Zhu[5,9]的團(tuán)隊(duì)提出并發(fā)展了高斯白噪聲、諧和噪聲等作用下單自由度或多自由度擬Hamilton系統(tǒng)的隨機(jī)平均理論和方法,解決了五類擬Hamilton系統(tǒng)平均方程的求解問(wèn)題.Xu[33]等建立了高斯色噪聲驅(qū)動(dòng)下一類隨機(jī)動(dòng)力學(xué)系統(tǒng)的平均原理及高斯白噪聲與色噪聲共同激勵(lì)下一類單自由度系統(tǒng)的隨機(jī)平均法[34].

近幾年來(lái),隨機(jī)平均法理論得到進(jìn)一步完善,并已被應(yīng)用于各類隨機(jī)動(dòng)力系統(tǒng)動(dòng)力學(xué)性質(zhì)的研究,其為研究更復(fù)雜的隨機(jī)動(dòng)力學(xué)系統(tǒng)和解決各種激勵(lì)下的隨機(jī)動(dòng)力學(xué)問(wèn)題提供了良好的方法.隨機(jī)平均的方法和理論也不再僅僅基于以高斯白噪聲為代表的不相關(guān)噪聲激勵(lì)下的隨機(jī)動(dòng)力學(xué)系統(tǒng),具有相關(guān)時(shí)間的分?jǐn)?shù)高斯噪聲激勵(lì)、非高斯列維噪聲及Markov切換的隨機(jī)動(dòng)力學(xué)系統(tǒng)(包含無(wú)窮維系統(tǒng))研究越來(lái)越引起學(xué)者的關(guān)注,取得了一定的發(fā)展.

本文根據(jù)國(guó)內(nèi)外研究現(xiàn)狀和發(fā)展趨勢(shì),綜述了基于非高斯列維噪聲、長(zhǎng)相關(guān)性分?jǐn)?shù)高斯噪聲及Markov切換的隨機(jī)復(fù)雜動(dòng)力學(xué)系統(tǒng)(包含無(wú)窮維系統(tǒng))隨機(jī)平均原理研究中的若干研究方向,并對(duì)存在的一些問(wèn)題以及進(jìn)一步的研究做了展望.

1 基于非高斯列維噪聲的隨機(jī)平均原理

在以往的大部分研究中,為了處理起來(lái)簡(jiǎn)便,研究人員考慮的都是高斯噪聲,它是布朗運(yùn)動(dòng)的形式導(dǎo)數(shù),一般用來(lái)描述連續(xù)型的微小的隨機(jī)因素.在大多數(shù)情況下,高斯的假設(shè)是比較合理的,它滿足中心極限定理,而且由于處理起來(lái)比較簡(jiǎn)單,理論推導(dǎo)比較容易,在許多領(lǐng)域都得到了廣泛的應(yīng)用.然而,高斯噪聲只是一種理想的噪聲源,它刻畫(huà)的是正常擴(kuò)散,即只能模擬均值在小范圍內(nèi)的起伏,而不能模擬大幅度的漲落.在實(shí)際應(yīng)用中,我們遇到的許多噪聲都是非高斯的,比如在生物醫(yī)學(xué)中的誘發(fā)電位噪聲、低頻的大氣噪聲以及各種其它人為噪聲等.這些噪聲的非高斯性使得它們具有更強(qiáng)的沖擊性,其所服從的分布比起正態(tài)分布,具有更多的尖峰與偶然性 (見(jiàn)圖1),而且其密度函數(shù)的拖尾與高斯密度函數(shù)相比,衰減的也更為緩慢 (見(jiàn)圖2)[35].這種情況下,以往基于高斯假定所得到的結(jié)論就需要被重新考慮,我們需要尋求一種更加廣義,能夠更好的與實(shí)際符合的分布,它的導(dǎo)數(shù)能更好地用來(lái)描述我們所遇到的噪聲.

圖1 不同的穩(wěn)定性指標(biāo)對(duì)應(yīng)的列維噪聲的概率密度函數(shù)Lα,β(ζ;D,μ)Fig.1 Probability density functions Lα,β(ζ;D,μ) for Lévy noise with different stability indexes

圖2 不同的偏斜參數(shù)對(duì)應(yīng)的列維噪聲的概率密度函數(shù)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]首先將隨機(jī)平均法運(yùn)用到泊松白噪聲激勵(lì)下的非線性系統(tǒng)的研究中,Zeng 和Zhu[37,40]研究了非高斯隨機(jī)激勵(lì)下非線性系統(tǒng)的隨機(jī)平均法.Xu[41]給出了非高斯列維噪聲驅(qū)動(dòng)下的隨機(jī)動(dòng)力系統(tǒng)的平均原理,Xu[42]還給出了在一類弱化的李普希茲條件下非高斯列維噪聲驅(qū)動(dòng)下的隨機(jī)動(dòng)力系統(tǒng)的平均原理,證明了平均后隨機(jī)動(dòng)力學(xué)系統(tǒng)的解依概率和均方收斂于原系統(tǒng)的解,給出了隨機(jī)平均法的理論依據(jù). Givon[43]根據(jù)快變量存在的不變測(cè)度,研究了兩尺度跳擴(kuò)散過(guò)程均方意義下的隨機(jī)平均原理,并得到相應(yīng)的收斂階為O(lnε)：

(3)

(4)

2 基于分?jǐn)?shù)高斯噪聲的隨機(jī)平均原理

在自然界等很多現(xiàn)象中的噪聲往往表現(xiàn)出相關(guān)性甚至是長(zhǎng)相關(guān)性的顯著特征,而分?jǐn)?shù)布朗運(yùn)動(dòng)為長(zhǎng)相關(guān)性噪聲的研究提供了重要的理論基礎(chǔ),它是一種比布朗運(yùn)動(dòng)更廣泛的隨機(jī)過(guò)程,具有的長(zhǎng)相關(guān)性、增量非獨(dú)立性已經(jīng)在金融[48,49]、地球物理學(xué)[50,51]、生物學(xué)[52,53]和腦功能信號(hào)分析[54,55]等方面有了一定的應(yīng)用.1968年,Mandelbrot和Van Ness[56]首先定義了“分?jǐn)?shù)布朗運(yùn)動(dòng)”,并給出分?jǐn)?shù)布朗運(yùn)動(dòng)的構(gòu)造.此后,分?jǐn)?shù)布朗運(yùn)動(dòng)驅(qū)動(dòng)隨機(jī)動(dòng)力系統(tǒng)的研究引起學(xué)者的關(guān)注.由于分?jǐn)?shù)布朗運(yùn)動(dòng)既不是半鞅又不是馬爾可夫過(guò)程,使得隨機(jī)積分這個(gè)完備的理論基礎(chǔ)并不適用于分?jǐn)?shù)布朗運(yùn)動(dòng)的研究.Xu[57-59]在前向路徑積分意義下,根據(jù)Khasminskii平均法,研究具有長(zhǎng)相關(guān)性分?jǐn)?shù)布朗運(yùn)動(dòng)的隨機(jī)平均原理,證明了具有長(zhǎng)相關(guān)性分?jǐn)?shù)布朗運(yùn)動(dòng)驅(qū)動(dòng)的動(dòng)力系統(tǒng)與平均后的隨機(jī)動(dòng)力系統(tǒng)在均方意義下是收斂的,并利用數(shù)值模擬的方法,驗(yàn)證了定理的正確性.Xu[60,61]還進(jìn)一步研究了分?jǐn)?shù)布朗運(yùn)動(dòng)驅(qū)動(dòng)的快-慢變系統(tǒng)的隨機(jī)平均原理.Deng和Zhu[62,64]根據(jù)分?jǐn)?shù)布朗運(yùn)動(dòng)驅(qū)動(dòng)的兩尺度隨機(jī)動(dòng)力系統(tǒng)隨機(jī)平均原理結(jié)果[60,61],提出并發(fā)展了分?jǐn)?shù)高斯噪聲等作用下單自由度或多自由度擬Hamilton系統(tǒng)的隨機(jī)平均理論和方法,解決了擬Hamilton系統(tǒng)平均方程的求解問(wèn)題.

3 基于Markov切換的隨機(jī)平均原理

1961年Krasovskii和Lidskii[65]首次提出Markov切換系統(tǒng).近些年來(lái),Markov切換系統(tǒng)在復(fù)雜網(wǎng)絡(luò)等非線性系統(tǒng)建模中起不可替代的作用,使得其迅速成為國(guó)內(nèi)外發(fā)展最活躍的前沿學(xué)科和研究熱點(diǎn)之一.所謂Markov切換系統(tǒng),即是以一個(gè)連續(xù)時(shí)間有限狀態(tài)的Markov鏈來(lái)控制系統(tǒng)的切換時(shí)刻和切換狀態(tài).由于Markov切換系統(tǒng)貼近應(yīng)用背景,數(shù)學(xué)描述清晰,可操作性強(qiáng),因而很快成為復(fù)雜網(wǎng)絡(luò)及其他非線性系統(tǒng)建模的重要參考依據(jù).事實(shí)上,自然界和人類社會(huì)中廣泛存在著的各種各樣的復(fù)雜系統(tǒng)都可以通過(guò)復(fù)雜網(wǎng)絡(luò)模型來(lái)描述.復(fù)雜網(wǎng)絡(luò)模型描述了復(fù)雜系統(tǒng)中元素之間、子系統(tǒng)之間、層次之間的相互作用以及系統(tǒng)與環(huán)境的相互作用.而當(dāng)系統(tǒng)元件出現(xiàn)突發(fā)故障或突然修復(fù)狀況,或突然出現(xiàn)外部干擾,以及子系統(tǒng)連接方式發(fā)生突變時(shí),復(fù)雜網(wǎng)絡(luò)系統(tǒng)很可能產(chǎn)生結(jié)構(gòu)或參數(shù)上的突發(fā)改變.例如種群系統(tǒng)中的環(huán)境噪音和地震等突發(fā)性現(xiàn)象對(duì)種群數(shù)量變化的影響；通訊系統(tǒng)中數(shù)據(jù)交換的障礙和機(jī)器故障對(duì)數(shù)據(jù)傳輸?shù)挠绊懀唤?jīng)濟(jì)金融系統(tǒng)中,國(guó)家宏觀調(diào)控對(duì)經(jīng)濟(jì)變化的影響等等.考慮到Markov切換系統(tǒng)在描述系統(tǒng)結(jié)構(gòu)或參數(shù)突然變化方面具有很大的優(yōu)勢(shì),因此,選用Markov切換系統(tǒng)來(lái)描述網(wǎng)絡(luò)的狀態(tài)更加貼合實(shí)際應(yīng)用背景.具有Markov切換的隨機(jī)動(dòng)力系統(tǒng)的隨機(jī)平均原理得到了廣泛的研究.Yin[66]研究了具有兩尺度Markov切換的跳擴(kuò)散模型的隨機(jī)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)代表連續(xù)時(shí)間兩尺度Markov切換.以及快變量慢變量耦合的Markov切換調(diào)制兩尺度隨機(jī)跳擴(kuò)散過(guò)程的隨機(jī)平均原理:

(6)

(7)

其中αε(t)代表連續(xù)時(shí)間兩尺度Markov切換,v(t)表示W(wǎng)iener過(guò)程. Bao,Yin和Yuan[68]考慮了加性α穩(wěn)定噪聲激勵(lì)的兩尺度隨機(jī)微分方程的隨機(jī)平均原理.宦榮華教授和朱位秋院士[69-70]等人給出了具有Markov切換的隨機(jī)平均法,并用此方法研究了Markov切換多自由度隨機(jī)擬不可積哈密頓系統(tǒng)的概率1穩(wěn)定性和在時(shí)滯反饋控制下Markov切換擬可積哈密頓系統(tǒng)的概率1穩(wěn)定性.

4 無(wú)窮維隨機(jī)動(dòng)力系統(tǒng)的隨機(jī)平均原理研究

5 結(jié)語(yǔ)

本文僅就我們關(guān)心的領(lǐng)域?qū)Ψ歉咚沽芯S噪聲、分?jǐn)?shù)高斯噪聲、Markov切換,及無(wú)窮維隨機(jī)系統(tǒng)隨機(jī)平均原理的研究做了介紹,還有許多方面沒(méi)有涉及.

目前來(lái)說(shuō),對(duì)于高斯白噪聲、列維噪聲激勵(lì)的有窮維系統(tǒng)的隨機(jī)平均原理較為系統(tǒng),已經(jīng)存在大量成熟的結(jié)果,而對(duì)于列維噪聲、分?jǐn)?shù)高斯噪聲激勵(lì)的無(wú)窮維系統(tǒng)及含Markov切換的隨機(jī)系統(tǒng)的隨機(jī)平均原理的研究尚在起步階段,研究還相當(dāng)?shù)厣?特別是針對(duì)于乘性列維噪聲、分?jǐn)?shù)高斯噪聲激勵(lì)的無(wú)窮維系統(tǒng)及含兩尺度Markov切換的隨機(jī)平均原理的研究是未來(lái)的重點(diǎn).

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*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

*國(guó)家自然科學(xué)基金資助項(xiàng)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.

? 通訊作者 E-mail:hsux3@nwpu.edu.cn