The persistence of synchronization under environmental noise

The pe sis ence o synch oniza ion unde en i onmen al noise By Tom´ as Ca aballo aand Pe e E. Kloeden b aDepa amen o de Ecuaciones Di e enciales y An´alisis Num´e ico, Uni e sidad de Se illa, Apdo. de Co eos 1160, 41080-Se illa, Spain E-mail: ca abal[email p o ec ed] bFachbe eich Ma hema ik, Johann Wol gang Goe he Uni e si ¨a D-60054 F ank u am Main, Ge many E-mail: kloe[email p o ec ed]ank u .de I is shown ha he synch oniza ion o dissipa i e sys ems pe sis s when hey a e dis u bed by addi i e noise no ma e how la ge he in ensi y o he noise p o ided asymp o ically s able s a iona y s ochas ic solu ions a e used ins ead o asymp o - ically s able equilib ia. Keywo ds: Synch oniza ion, addi i e noise, andom a ac o , s a iona y s ochas ic p ocess, one-sided Lipschi z dissipa i e condi ion 1. In oduc ion Synch oniza ion o coupled sys ems is a e y well known phenomenon in biology and physics, and also in he social sciences. A eadable desc ip i e accoun o i s di e si y o occu ence can be ound in he ecen book o S oga z (2003), which con ains an ex ensi e lis o e e ences. The synch oniza ion o coupled dissipa i e sys ems has been in es iga ed ma hema ically in he case o au onomous sys ems by A aimo ich and Rod igues (1998), Ca alho e al. (1998) and Rod igues (1996), bo h o asymp o ically s able equilib ia and gene al a ac o s, such as chao ic a - ac o s. Analogous esul s also hold o nonau onomous sys ems (Kloeden (2003)), bu equi e a new concep o a nonau onomous a ac o . In his no e we in es iga e he e ec o addi i e noise on he synch oniza ion o coupled dissipa i e sys ems wi h asymp o ically s able equilib ia, which esul s in a coupled sys em o I o s ochas ic di e en ial equa ions. Such noise is o en conside ed as modelling backg ound en i onmen al e ec s. We show ha synch o- niza ion pe sis s independen ly o noise in ensi y in e ms o asymp o ically s able s a iona y s ochas ic solu ions a he han equilib ia. Such asymp o ically s able s a iona y s ochas ic solu ions a e a special case o andom a ac o s (see C auel & Flandoli (1994), C auel e al. (1997), Schmal uss (1992)) which a e a p obabilis ic coun e pa o he de e minis ic nonau onomous a ac o used in Kloeden (2003). We p esen some equi ed backg ound ma e ial in Sec ion 2 and hen o mula e he p oblem o synch oniza ion o dissipa i e sys ems o bo h he de e minis ic and s ochas ic cases in Sec ion 3. In Sec ion 4 we show ha he uncoupled sys ems wi h addi i e noise ha e andom a ac o s consis ing o single s a iona y s ochas ic A icle submi ed o Royal Socie y T EX Pape 2T. Ca aballo & P.E. Kloeden p ocesses and hen es ablish he co esponding esul o coupled sys ems in Sec ion 5. Finally, he asymp o ic beha iou o he coupled sys ems in he limi as he coupling pa ame e becomes a bi a ily la ge a e p esen ed in Sec ion 6. 2. Backg ound ma e ial We need some backg ound no a ion and esul s. Le (Ω,F,P) be a p obabili y space. Following A nold (1998) a andom dynamical sys em (RDS) (θ, φ) on Ω ×Rd consis s o a me ic dynamical sys em θon Ω and a cocycle mapping φ:R+×Ω×Rd →Rd. Essen ially (and su icien o ou pu poses he e), θ ep esen s he d i ing noise p ocess and φ he s a e space e olu ion o he sys em. Fo example, o a s ochas ic di e en ial equa ion on Rdwi h a wo-sided scala Wiene p ocess W , i.e. de ined o all ∈R,θis de ined by θ ω(·) = ω( +·)−ω(·) on a canonical sample space Ω = C0(R,R) whe e, by de ini ion, W (ω) := ω( ), ∈R, and φis de ined by φ( , ω, x0) = Xx0 (ω), he solu ion o he SDE s a ing a Xx0 0(ω) = x0. See he exposi ions in A nold (1998) and Kloeden e al. (1999) o mo e de ails. A amily b A={A(ω), ω ∈Ω}o nonemp y measu able compac subse s A(ω) o Rdis called φ-in a ian i φ( , ω, A(ω)) = A(θ ω) o all ≥0 and is called a andom a ac o i in addi ion i is pa hwise pullback a ac ing in he sense ha H∗ d(φ( , θ− ω, D(θ− ω)), A(ω)) →0 as → −∞ o all sui able (i.e. in a gi en a ac ing uni e se as, o ins ance, in Kloeden e al. (1999)) amilies o b D={D(ω), ω ∈Ω}o nonemp y measu able bounded subse s D(ω) o Rd. He e H∗ dis he Hausdo semi-dis ance on Rd. The ollowing esul (c . A nold (1998), Kloeden e al. (1999), Schmal uss (1992)) ensu es he exis ence o a andom a ac o . Theo em 2.1. Le (θ, φ)be an RDS on Ω×Rd. I he e exis s a amily b B= {B(ω), ω ∈Ω}o nonemp y measu able compac subse s B(ω)o Rdand a Tb D,ω ≥ 0such ha φ( , θ− ω, D(θ− ω)) ⊂B(ω),∀ ≥Tb D,ω o all amilies b D={D(ω), ω ∈Ω}in he gi en a ac ing uni e se, hen he RDS (θ, φ)has a andom a ac o b A={A(ω), ω ∈Ω}wi h he componen subse s de ined o each ω∈Ωby A(ω) = s>0[ ≥s φ( , θ− ω, B(θ− ω)). No e ha i he andom a ac o consis s o single on se s, i.e A(ω) = {X∗(ω)} o some andom a iable X∗, hen X∗ (ω) := X∗(θ ω) is a s a iona y s ochas ic p ocess. We also need he ollowing lemma a. Lemma 2.2. Le {xn}be a sequence in a comple e me ic space (X, d)such ha e e y subsequence {xni}has a subsequence {xnij}con e ging o a common limi x∗. Then he sequence {xn}con e ges o x∗. A icle submi ed o Royal Socie y Pe sis ence o synch oniza ion unde en i onmen al noise 3 P oo . I no , hen he e would exis an η > 0 and a subsequence {xni}such ha d(xni, x∗)≥η o all i, which is no possible because {xni}has a subsequence {xnij}con e ging o he x∗. Lemma 2.3. Le W be a wo-sided Wiene p ocess, i.e. de ined o all ∈R. Then he in eg als νR −∞ e−ν( −s)dW (ω)a e pa hwise uni o mly bounded in ν > 0 on ini e ime in e als [T1, T2]o Rand he in eg als Z T1 e−ν( −s)dW (ω)→0as ν→ ∞ pa hwise on ini e ime in e als [T1, T2]o R. P oo . In eg a ing by pa s (see Mao (1997)), we ha e Z −∞ e−ν( −s)dWs(ω) = W (ω)−νZ −∞ e−ν( −s)Ws(ω)ds. Then we use he sub-exponen ial g ow h o he pa hs o Wiene p ocesses. Fo he second pa we i s no ice ha νZ T1 e−ν( −s)W (ω)ds =W (ω)−e−ν( −T1)W (ω) o any ω∈Ω. In eg a ing again by pa s i ollows Z T1 e−ν( −s)dWs(ω) = W (ω)−e−ν( −T1)WT1(ω)−νZ T1 e−ν( −s)Ws(ω)ds =e−ν( −T1)(W (ω)−WT1(ω)) +νZ T1 e−ν( −s)(W (ω)−Ws(ω)) ds om which he esul ollows. 3. Fo mula ion o he p oblem Suppose we ha e wo au onomous o dina y di e en ial equa ions in Rd, dx d = (x),dy d =g(y),(3.1) which a e su icien ly egula o ensu e he o wa ds exis ence and uniqueness o solu ions and sa is y one-sided dissipa i e Lipschi z condi ions hx1−x2, (x1)− (x2)i ≤ −L|x1−x2|2, hy1−y2, g(y1)−g(y2)i ≤ −L|y1−y2|2,(3.2) on Rd o some L > 0, and hus ha e unique equilib ia ¯xand ¯y, espec i ely, which a e globally asymp o ically s able (c . Kloeden (2004)). No ice ha he con inui y o and g, and he one-sided dissipa i e Lipschi z condi ions (3.2) ensu e he o wa ds exis ence and uniqueness o solu ions o (3.1). A icle submi ed o Royal Socie y 4T. Ca aballo & P.E. Kloeden Conside now he dissipa i ely coupled sys em dx d = (x) + ν(y−x),dy d =g(y) + ν(x−y) (3.3) wi h ν > 0. I can be shown (see A aimo ich and Rod igues (1998), and Ca alho e al. (1998)) ha his also has a unique equilib ium (¯xν,¯yν), which is globally asymp o ically s able. Mo eo e , (¯xν,¯yν)→(¯z, ¯z) as ν→ ∞, whe e ¯zis he unique globally asymp o ically s able equilib ium o he “a e aged” sys em dz d =1 2( (z) + g(z)) .(3.4) This phenomena is known as synch oniza ion. Analogous esul s hold o mo e gen- e al au onomous a ac o s (c . A aimo ich and Rod igues (1998), Ca alho e al. (1998)) as well as o nonau onomous sys ems (Kloeden (2003)) wi h app op ia ely de ined nonau onomous a ac o s. The aim o his no e is o show ha his synch oniza ion e ec is p ese ed un- de addi i e noise p o ided equilib ia a e eplaced by s a iona y andom solu ions. Speci ically, we conside wo I o s ochas ic di e en ial equa ions in Rd, dX = (X )d +α dW1 , dY =g(Y )d +β dW 2 ,(3.5) whe e α,β∈Rd +a e cons an ec o s wi h no componen s equal o ze o, W1 , W2 a e independen wo-sided scala Wiene p ocesses†, and ,ga e as abo e, in pa icula , sa is ying he one-sided dissipa i e Lipschi z condi ions (3.2). We will show in he nex sec ion ha each o hese s ochas ic sys ems has a pa hwise asymp o ically s able andom a ac o consis ing o a s a iona y andom a iable. Fo example, o linea d i e ms, i.e he SDEs dX =−X d +α dW1 , dY =−Y d +β dW 2 ,(3.6) hese andom a iables a e gi en explici ly by ¯ X =αe− Z −∞ esdW1 s,¯ Y =βe− Z −∞ esdW2 s.(3.7) The synch onized sys em co esponding o SDEs (3.5) eads dX = ( (X ) + ν(Y −X )) d +α dW1 , dY = (g(Y ) + ν(X −Y )) d +β dW 2 .(3.8) I will be shown ha his sys em is dissipa i e and has a unique s a iona y solu ion (¯ Xν ,¯ Yν ), which is pa hwise globally asymp o ically s able wi h (¯ Xν ,¯ Yν )→(¯ Z∞ ,¯ Z∞ ) as ν→ ∞, pa hwise on ini e ime in e als [T1, T2] o R, whe e ¯ Z∞ is he unique pa hwise globally asymp o ically s able s a iona y solu ion o he “a e aged” SDE dZ =1 2( (Z ) + g(Z )) d +1 2α dW 1 +1 2β dW2 .(3.9) †al e na i ely, one could ake α,βscala alued and W1 ,W2 ec o alued A icle submi ed o Royal Socie y Pe sis ence o synch oniza ion unde en i onmen al noise 5 Fo example, o he linea SDEs (3.6) we ha e dZ =−Z d +1 2α dW1 +1 2β dW2 wi h ¯ Z∞ =1 2e− µαZ −∞ esdW1 s+βZ −∞ esdW2 s¶=1 2¡¯ X +¯ Y ¢, his a e aged exp ession being due o he special linea s uc u e. In addi ion o he one-sided Lipschi z dissipa i e condi ion on he unc ions and gwe assume he ollowing in eg abili y condi ion: The e exis s m0>0 such ha o any m∈(0, m0], and any con inuous unc ion u:R→Rdwi h sub-exponen ial g ow h i ollows Z −∞ ems| (u(s))|2ds < +∞,Z −∞ ems|g(u(s))|2ds < +∞.(3.10) Wi hou loss o gene ali y, we can assume ha L≤m0. 4. The uncoupled sys ems wi h addi i e noise We conside he i s o he uncoupled equa ions in (3.5), dX = (X )d +α dW 1 .(4.1) I s solu ion pa hs a e gene ally no di e en iable, so in o de o use he one-sided dissipa i e Lipschi z condi ion (3.2) we conside he di e ence X −¯ X whe e ¯ X is he O ns ein-Uhlenbeck s a iona y p ocess (3.7) sa is ying he i s o he linea equa ions (3.6). This di e ence is pa hwise di e en iable since he pa hs X and ¯ X a e con inuous and sa is y he in eg al equa ion X −¯ X =X0−¯ X0+Z 0¡ (Xs) + ¯ Xs¢ds, which, by he undamen al heo em o calculus, is hus equi alen o he di e en ial exp ession d d ¡X −¯ X ¢= (X ) + ¯ X , om which i ollows ha d d ¯¯X −¯ X ¯¯2= 2 X −¯ X , (X )− (¯ X )®+ 2 X −¯ X , (¯ X ) + ¯ X )® ≤ −2L¯¯X −¯ X ¯¯2+L¯¯X −¯ X ¯¯2+1 L¯¯ (¯ X ) + ¯ X )¯¯2 and hus ¯¯X −¯ X ¯¯2≤¯¯X 0−¯ X 0¯¯2e−L( − 0)+e−L LZ 0 eLs ¯¯ (¯ Xs) + ¯ Xs¯¯2ds. A icle submi ed o Royal Socie y 6T. Ca aballo & P.E. Kloeden Pa hwise pullback con e gence (i.e. as 0→ −∞) gi es pullback abso p ion (c . Theo em 2.1) ¯¯X −¯ X ¯¯2≤R2 X(θ ω) := 1 + e−L LZ −∞ eLs ¯¯ (¯ X(θsω)) + ¯ X(θsω)¯¯2ds o all ≥Tb D(ω) o app op ia e amilies b D(ω) o bounded se s {D(θ ω), ∈R}o ini ial condi ions. The in eg als he e exis due o assump ion (3.10) and he ac ha O ns ein- Uhlenbeck p ocesses inhe i he sub-exponen ial g ow h o hei Wiene p ocesses. Thus ¯¯X (ω)−¯ X (ω)¯¯≤RX(θ ω),∀ ≥Tb D(ω), and so |X (ω)| ≤ ¯¯¯ X (ω)¯¯+RX(θ ω),∀ ≥Tb D(ω), which means ha his sys em has a andom a ac o b A={A(ω), ω ∈Ω}. Bu he di e ence o any wo solu ions sa is ies he di e en ial inequali y d d ¯¯X1 −X2 ¯¯2≤ −2L¯¯X1 −X2 ¯¯2, which means all solu ions con e ge pa hwise o each o he and hus he andom a - ac o se s a e single on se s A(ω) = {X∗(ω)}, i.e. he andom a ac o is o med by a s a iona y andom p ocess X∗ (ω) = X∗(θ ω) which pa hwise a ac s all o he solu ions. An analogous si ua ion holds o he second equa ion o he uncoupled equa ions in (3.5). 5. The synch onized sys em wi h addi i e noise To show ha he synch onized sys em (3.8) is s ongly dissipa i e, we use he OU p ocesses X∗,ν =αe−ν Z −∞ eνs dW 1 s, Y ∗,ν =βe−ν Z −∞ eνs dW 2 s,(5.1) which a e he s a iona y solu ions o he linea equa ions dX =−νX d +α dW 1 , dY =−νY d +β dW 2 .(5.2) The di e ences o he solu ions o (3.8) and hese s a iona y solu ions a e pa hwise di e en iable and sa is y he sys em o andom di e en ial equa ions d d ¡X −X∗,ν ¢= (X ) + ν(Y −X ) + νX∗,ν d d ¡Y −Y∗,ν ¢=g(Y ) + ν(X −Y ) + νY ∗,ν , A icle submi ed o Royal Socie y Pe sis ence o synch oniza ion unde en i onmen al noise 7 which, wi h Uν := X −X∗,ν and Vν := Y −Y∗,ν , is equi alen o d d Uν = (Uν +X∗,ν )+ν(Vν−Uν)+νY ∗,ν ,d d Vν=g(Y )+ν(Uν−Vν)+νX∗,ν . Thus 1 2 d d ¡|Uν |2+|Vν |2¢=Uν , (Uν +X∗,ν )− (X∗,ν )®+Vν , g(Vν +Y∗,ν )−g(Y∗,ν )® +Uν , (X∗,ν ) + νY ∗,ν ®+Vν , g(Y∗,ν ) + νX∗,ν ® −νhUν −Vν , Uν −Vν i ≤ −L¡|Uν |2+|Vν |2¢+L 2¡|Uν |2+|Vν |2¢ +2 L¯¯ (X∗,ν ) + νY ∗,ν ¯¯2+2 L¯¯g(Y∗,ν ) + νX∗,ν ¯¯2, and hence d d ¡|Uν |2+|Vν |2¢≤ −L¡|Uν |2+|Vν |2¢+4 L¯¯ (X∗,ν ) + νY ∗,ν ¯¯2+4 L¯¯g(Y∗,ν ) + νX∗,ν ¯¯2. This means ha |Uν (ω)|2+|Vν (ω)|2is pa hwise abso bed by he amily b Bν= {Bν(ω), ω ∈Ω}o closed balls in R2dcen e ed on he o igin and o adius Rν(ω), whe e R2 ν(ω) is de ined by 1+ 4 LZ0 −∞ eLs ³| (X∗,ν (θsω) + νY ∗,ν (θsω)|2ds +|g(Y∗,ν (θsω) + νX∗,ν (θsω)|2´ds. Hence, by Theo em 2.1, he synch onized sys em has a andom a ac o b Aν= {Aν(ω), ω ∈Ω}wi h Aν(ω)⊂Bν(ω). Bu he di e ence (∆X ,∆Y ) = (X1 − X2 , Y 1 −Y2 ) o any pai o solu ions sa is ies he sys em o andom di e en ial equa ions d d ∆X = (X1 )− (X2 )+ν(∆Y −∆X ),d d ∆Y =g(Y1 )−g(Y2 )+ν(∆X −∆Y ), so 1 2 d d ¡|∆X |2+|∆Y |2¢=∆X , (X1 )− (X2 )®+∆Y , g(Y1 )−g(Y2 )® −νh∆X −∆Y ,∆X −∆Y i ≤ −L¡|∆X |2+|∆Y |2¢, om which we ob ain |∆X (ω)|2+|∆Y (ω)|2≤¡|∆X0(ω)|2+|∆Y0(ω)|2¢e−2L , which means all solu ions con e ge pa hwise o each o he as → ∞. Thus he andom a ac o consis s o single on se s o med by an o de ed pai o s a iona y p ocesses ( ¯ Xν (ω),¯ Yν (ω)). A icle submi ed o Royal Socie y 8T. Ca aballo & P.E. Kloeden 6. The synch onized s a iona y solu ions as ν→ ∞ Lemma 6.1. ¯ Xν (ω)−¯ Yν (ω)→0as ν→ ∞ pa hwise on any bounded ime in e al [T1, T2]o R. P oo . Sub ac ing he second om he i s equa ion o (3.8) gi es d¡¯ Xν −¯ Yν ¢=¡−2ν¡¯ Xν −¯ Yν ¢+ (¯ Xν )−g(¯ Yν )¢d +α dW1 −β dW 2 , o , wi h Dν =¯ Xν −¯ Yν , d¡Dν e2ν ¢=e2ν ¡ (¯ Xν )−g(¯ Yν )¢d +αe2ν dW1 −βe2ν dW 2 , so pa hwise |Dν | ≤ e−2νT1|Dν T1|+Z T1 e−2ν( −s)¡¯¯ (¯ Xν s)¯¯+¯¯g(¯ Yν s)¯¯¢ds +|α|¯¯¯¯Z T1 e−2ν( −s)dW1 ¯¯¯¯+|β|¯¯¯¯Z T1 e−2ν( −s)dW2 ¯¯¯¯.(6.1) By Lemma 2.3 we see ha he adius Rν(θ ω) is pa hwise uni o mly bounded on each bounded ime in e al [T1, T2], so |Xν (ω)|,|Yν (ω))|and |Dν 0(ω))|a e pa hwise uni o mly bounded on each bounded ime in e al [T1, T2]. Then by Lemma 2.3 and condi ion (3.10) we see ha all o he in eg als in (6.1) con e ge o ze o as ν→ ∞ pa hwise on he bounded ime in e al [T1, T2]. Theo em 6.2. (¯ Xνn ,¯ Yνn )→(Z∞ , Z∞ )pa hwise uni o mly on bounded ime in- e als [T1, T2] o any sequence νn→ ∞, whe e Z∞ is he s a iona y s ochas ic solu ion o he a e aged SDE dZ =1 2( (Z ) + g(Z )) d +1 2αdW 1 +1 2βdW2 .(6.2) P oo . De ine Zν := 1 2¡¯ Xν +¯ Yν ¢∀ ∈R and obse e ha Zν sa is ies he equa ion dZν =1 2¡ (¯ Xν ) + g(¯ Yν )¢d +1 2α dW1 +1 2β dW2 . Also de ine ¯ Z := 1 2¡¯ X +¯ Y ¢∀ ∈R whe e ¯ X and ¯ Y a e he OU p ocesses sa is ying he linea SDEs (3.6). The di e ence Zν −¯ Z is pa hwise con inuous and hus by Lemma 2.3 and he de ini ions o he espec i e abso bing se s, equi-bounded w. . . ν > 0 on he bounded ime in e al [T1, T2]. Mo eo e Zν −¯ Z is pa hwise di e en iable and sa is ies he andom di e en ial exp ession d d ¡Zν −¯ Z ¢=1 2 (¯ Xν ) + 1 2g(¯ Yν ) + 1 2¯ X +1 2¯ Y , A icle submi ed o Royal Socie y Pe sis ence o synch oniza ion unde en i onmen al noise 9 Since ¯¯¯¯ d d ¡Zν (ω)−¯ Z (ω)¢¯¯¯¯≤1 2¯¯ (¯ Xν (ω))¯¯+1 2¯¯g(¯ Yν (ω))¯¯+1 2¯¯¯ X (ω)¯¯+1 2¯¯¯ Y (ω)¯¯ ≤MT1,T2(ω)<∞ by Lemma 2.3, we can use he Ascoli Theo em o conclude ha o any sequence νn → ∞, he e is a (possibly) andom subsequence νnj(ω)→ ∞ such ha Zνnj (ω)− ¯ Z (ω)→Z∞ (ω)−¯ Z (ω) as j→ ∞, and hus ha Zνnj (ω)→Z∞ (ω) as j→ ∞. Now Zνnj (ω)−¯ Yνnj (ω) = 1 2³¯ Xνnj (ω)−¯ Yνnj (ω)´→0, Zνnj (ω)−¯ Xνnj (ω) = 1 2³¯ Yνnj (ω)−¯ Xνnj (ω)´→0 as νnj→ ∞, so ¯ Xνnj (ω)=2Zνnj (ω)−¯ Yνnj (ω)→Z∞ (ω), ¯ Yνnj (ω)=2Zνnj (ω)−¯ Xνnj (ω)→Z∞ (ω) as νnj→ ∞. Mo eo e , Zν −¯ Z =Zν −¯ Z +1 2Z T1 (¯ Xν s)ds +1 2Z T1 g(¯ Yν s)ds +1 2Z T1 ¯ Xsds +1 2Z T1 ¯ Ysds which con e ges pa hwise o Z∞ =Z∞ T1+1 2Z T1 (Z∞ s)ds +1 2Z T1 g(Z∞ s)ds +¯ Z −¯ ZT1+1 2Z T1 ¯ Xsds +1 2Z T1 ¯ Ysds =Z∞ T1+1 2Z T1 (Z∞ s)ds +1 2Z T1 g(Z∞ s)ds +1 2αZ T1 dW1 s+1 2βZ T1 dW2 s, on he in e al [T1, T2], so Z∞ is a solu ion o he SDE (6.2) o all ∈R. The d i o his SDE sa is ies he s ongly dissipa i e one-sided Lipschi z condi ion (3.2), so i has a andom a ac o consis ing o a single on se o med by a s a iona y s ochas ic p ocess which hus mus be equal o Z∞ . Finally, we no e ha pa hwise all possible subsequences he e ha e he same limi , so by Lemma 2.2 e e y ull sequence Zνn ac ually con e ges o Z∞ as νn→ ∞. As a s aigh o wa d consequence o he a gumen s in he p e ious p oo we ha e A icle submi ed o Royal Socie y