Pathwise description of dynamic pitchfork bifurcations with additive noise

Transcription

1 Pahwie decripion of dynamic pichfork bifurcaion wih addiive noie Nil Berglund and Barbara Genz Abrac The low drif (wih peed ) of a parameer hrough a pichfork bifurcaion poin, known a he dynamic pichfork bifurcaion, i characerized by a ignifican delay of he raniion from he unable o he able ae. We decribe he effec of an addiive noie, of ineniy σ, by giving precie eimae on he behaviour of he individual pah. We how ha unil ime afer he bifurcaion, he pah are concenraed in a region of ize σ/ /4 around he bifurcaing equilibrium. Wih high probabiliy, hey leave a neighbourhood of hi equilibrium during a ime inerval [, c log σ ], afer which hey are likely o ay cloe o he correponding deerminiic oluion. We derive exponenially mall upper bound for he probabiliy of he e of excepional pah, wih explici value for he exponen. Dae. Augu 4, 2. Revied. April 9, 2. 2 Mahemaic Subjec Claificaion. 37H2, 6H (primary), 34E5, 93E3 (econdary). Keyword and phrae. Dynamic bifurcaion, pichfork bifurcaion, addiive noie, bifurcaion delay, ingular perurbaion, ochaic differenial equaion, dynamical yem, pahwie decripion, concenraion of meaure. Inroducion Phyical yem are ofen decribed by ordinary differenial equaion (ODE) of he form dx d = f(x, λ), (.) where x i he ae of he yem, λ a parameer, and denoe ime. The model (.) may however be oo crude, ince i neglec all kind of perurbaion acing on he yem. We are inereed here in he combined effec of wo perurbaion: a low drif of he parameer, and an addiive noie. A lowly drifing parameer λ =, (wih ), may model he deerminiic change in ime of ome exerior influence, uch a he climae acing on an ecoyem or a magneic field acing on a ferromagne. Obviouly, nonrivial dynamic can only be expeced when λ i allowed o vary by an amoun of order, and hu he yem ha o be conidered on he ime cale. Thi i uually done by inroducing he low ime =, which ranform (.) ino he ingularly perurbed equaion dx d = f(x, ). (.2) I i known ha oluion of hi yem end o ay cloe o able equilibrium branche of f [Gr, Ti], ee Fig. a. New, and omeime urpriing phenomena occur when uch an

2 µ µ Ü Øµ Ü Ø Ø Ü Ø Ø Ü Øµ Ü Øµ Figure. Soluion of he lowly ime-dependen equaion (.2) repreened in he (, x)- plane. (a) Sable cae: A able equilibrium branch x () arac nearby oluion x de. Two oluion wih differen iniial condiion are hown. They converge exponenially fa o each oher, a well a o a neighbourhood of order of x (). (b) Pichfork bifurcaion: The able equilibrium x = become unable a = (broken line) and expel wo able equilibrium branche ±x (). A oluion x de i hown, which i araced by x =, and ay cloe o he origin for a finie ime afer he bifurcaion. Thi phenomenon i known a bifurcaion delay. ÙÖ ½º Æ Ð Ö ÐÙÒ Ò Ö Ö ÒØÞ ÝÒ Ñ Ô Ø ÓÖ ÙÖ Ø ÓÒ Û Ø Ø Ú ÒÓ equilibrium branch undergoe a bifurcaion. Thee phenomena are uually called dynamic bifurcaion [Ben]. In he cae of he Hopf bifurcaion, when he equilibrium ge unable while expelling a able periodic orbi, he bifurcaion i ubanially delayed: oluion of (.2) rack he unable equilibrium (for a non-vanihing ime inerval in he limi ) before jumping o he limi cycle [Sh, Ne]. A imilar phenomenon exi for he dynamic pichfork bifurcaion of an equilibrium wihou drif, he imple example being f(x, ) = x x 3 (Fig. b). The delay ha been oberved experimenally, for inance, in laer [ME] and in a damped roaing pendulum [BK]. Thee phenomena have he advanage of providing a genuinely dynamic poin of view for he concep of a bifurcaion. Alhough one ofen ay ha a bifurcaion diagram (repreening he aympoic ae of he yem a a funcion of he parameer) i obained by varying he conrol parameer λ, he impaien experimenali aking hi lierally may have he urprie o dicover unable aionary ae of he yem ()he inveigae. The aympoic ae of he yem (.) wih lowly varying parameer λ() = λ() may depend no only on he iniial condiion (x, ½ ), bu alo on he hiory of variaion of he parameer λ()}. The perurbaion of (.) by addiive noie can be modeled by a ochaic differenial equaion (SDE) of he form dx = f(x, λ) d + σ dw, (.3) where W denoe he andard Wiener proce, and σ meaure he noie ineniy. A widepread approach i o analye he probabiliy deniy of x, which aifie he Fokker Planck equaion. In paricular, if f can be wrien a he gradien of a poenial funcion F, hen here i a unique aionary deniy p(x, λ) = e F (x,λ)/σ2 /N, where N i he normalizaion. Thi formula how ha for mall noie ineniy, he aionary deniy i harply peaked around able equilibria of f. Unforunaely, he erm dynamical bifurcaion i ued in a differen ene in he conex of random dynamical yem, namely o decribe a bifurcaion of he family of invarian meaure a oppoed o a phenomenological bifurcaion, ee for inance [Ar]. 2

3 Tha mehod ha, however, wo major limiaion. The fir one i ha he Fokker Planck equaion i difficul o olve, excep in he linear and in he gradien cae. The econd limiaion i more eriou: he deniy give no informaion on correlaion in ime, and even when he deniy i rongly localized, individual pah can perform large excurion. Thi i why oher approache are imporan. A claical one i baed on he compuaion of fir exi ime from he neighbourhood of able equilibria [FW, FJ]. The effec of bifurcaion ha been udied more recenly by mehod baed on he concep of random aracor [CF94, Schm, Ar]. In paricular, Crauel and Flandoli howed ha according o heir definiion, Addiive noie deroy a pichfork bifurcaion [CF98]. The phyical inerpreaion of random aracor i, however, no raighforward, and alernaive characerizaion of ochaic bifurcaion are deirable. In he ame way a lowly varying parameer help our underanding of bifurcaion in he deerminiic cae, i can provide a new poin of view in he cae of random dynamical yem. Le u conider he combined effec of a lowly drifing parameer and addiive noie on he ODE (.). We will focu on he cae of a pichfork bifurcaion, where he queion How doe he addiive noie affec he bifurcaion delay? and Where doe he pah go afer croing he bifurcaion poin? are of major phyical inere. The iuaion of he drif erm f in (.3) depending explicily on ime i coniderably more difficul han he auonomou cae, and hu much le underood. One can expec, however, ha a low ime dependence make he problem acceible o perurbaion heory, and ha one may ake advanage of echnique developed o udy ingularly perurbed equaion uch a (.2). Wih λ =, Equaion (.3) become dx = f(x, ) d + σ dw. (.4) If we inroduce again he low ime =, he Brownian moion i recaled, reuling in he SDE dx = f(x, ) d + σ dw. (.5) Our analyi of (.5) i rericed o one-dimenional x. The noie ineniy σ hould be conidered a a funcion of. Indeed, ince we now conider he equaion on he ime cale, a conan noie ineniy would lead o an infinie preading of rajecorie a. In he cae of he pichfork bifurcaion, we will need o aume ha σ. Variou paricular cae of equaion (.5) have been udied before, from a mahemaically non-rigorou poin of view. In he linear cae f(x, λ) = λx, he diribuion of fir exi ime wa inveigaed and compared wih experimen in [TM, SMC, SHA], while [JL] derived a formula for he la croing of zero. In he cae f(x, λ) = λx x 3, [Ga] udied he dependence of he delay on and σ numerically, while [Ku] conidered he aociaed Fokker Planck equaion, he oluion of which he approximaed by a Gauian anaz. In he preen work, we analye (.5) for a general cla of odd funcion f(x, λ) undergoing a pichfork bifurcaion. We ue a differen approach, baed on a precie conrol of he whole pah x } of he proce. The reul hu conain much more informaion han he probabiliy deniy. I alo urn ou ha he echnique we ue allow o deal wih nonlineariie in quie a naural way. Our reul can be ummarized in he following way (ee Fig. 2): Soluion of he deerminiic equaion (.2) aring near a able equilibrium branch of f are known o reach a neighbourhood of order of ha branch in a ime of order log. We how ha he pah of he SDE (.5) wih he ame iniial condiion 3

4 Ü µ Ü Ø Ø Ü Øµ Ü Øµ µ Ü Ø Ø ¼ Ô Ø Figure 2. A ypical pah x of he ochaic differenial equaion (.5) near a pichfork bifurcaion. We prove ha wih probabiliy exponenially cloe o, he pah ha he following behaviour. For, i ay in a rip B(h) conruced around he deerminiic oluion wih he ame iniial condiion. Afer =, i leave he domain D a a random ime τ = τ D, which i ypically of he order log σ. Then i ay (up o ime of order a lea) in a rip A τ (h) conruced around he deerminiic oluion x de,τ aring a ime τ on he boundary of D. The widh of B(h) and A τ (h) are proporional o a parameer h aifying σ h. ÙÖ ¾º Æ Ð Ö ÐÙÒ Ò Ö Ö ÒØÞ ÝÒ Ñ Ô Ø ÓÖ ÙÖ Ø ÓÒ Û Ø Ø Ú ÒÓ are ypically concenraed in a neighbourhood of order σ of he deerminiic oluion (Theorem 2.4). A paricular oluion of he deerminiic equaion (.2) i known o exi in a neighbourhood of order of each unable equilibrium branch of f. Pah ha ar in a neighbourhood of order σ of hi oluion are likely o leave ha neighbourhood in a ime of order log (Theorem 2.6). When a pichfork bifurcaion occur a x =, =, he ypical pah are concenraed in a neighbourhood of order σ/ /4 of he deerminiic oluion wih he ame iniial condiion up o ime (Theorem 2.). Afer he bifurcaion poin, he pah are likely o leave a neighbourhood of order of he unable equilibrium before a ime of order log σ (Theorem 2.). ½ Once hey have lef hi neighbourhood, he pah remain wih high probabiliy in a region of ize σ/ around he correponding deerminiic oluion, which approache a able equilibrium branch of f like / 3/2 (Theorem 2.2). Thee reul how ha he bifurcaion delay, which i oberved in he dynamical yem (.2), i deroyed by addiive noie a oon a he noie i no exponenially mall. Do hey mean ha he dynamic bifurcaion ielf i deroyed by addiive noie? Thi i mainly a maer of definiion. On he one hand, we will ee ha independenly of he iniial condiion, he probabiliy of reaching he upper, raher han he lower branch emerging from he bifurcaion poin, i cloe o 2. The aympoic ae i hu eleced by he noie, and no by he iniial condiion. Hence, he bifurcaion i deroyed in he ene of [CF98]. On he oher hand, individual pah are concenraed near he able 4

5 equilibrium branche of f, which mean ha he bifurcaion diagram will be made viible by he noie, much more o han in he deerminiic cae. So we do oberve a qualiaive change in behaviour when λ change i ign, which can be conidered a a bifurcaion. The precie aemen and a dicuion of heir conequence are given in Secion 2. In Secion 2.2, we analye he moion near equilibrium branche away from bifurcaion poin. The acual pichfork bifurcaion i dicued in Secion 2.3. A few conequence are derived in Secion 2.4. Secion 3 conain he proof of he fir wo heorem on he moion near nonbifurcaing equilibria, while he proof of he la hree heorem on he pichfork bifurcaion are given in Secion 4. Acknowledgemen: I a grea pleaure o hank Anon Bovier for haring our enhuiam. We enjoyed lively dicuion and hi conan inere in he progre of our work. The cenral idea were developed during muual vii in Berlin rep. Alana. N. B. hank he WIAS and B. G. hank Turgay Uzer and he School of Phyic a Georgia Tech for heir kind hopialiy. N. B. wa parially uppored by he Fond Naional Suie de la Recherche Scienifique, and by he Nonlinear Conrol Nework of he European Communiy, Gran ERB FMRXCT Saemen of reul 2. Preliminarie We conider nonlinear SDE of he form dx = f(x, ) d + σ dw, x = x, (2.) where W } i a andard Wiener proce on ome probabiliy pace (Ω, F, P). Iniial condiion x are alway aumed o be quare-inegrable wih repec o P and independen of W }. All ochaic inegral are conidered a Iô inegral, bu noe ha Iô and Sraonovich inegral agree for inegrand depending only on ime and ω. Wihou furher menioning we alway aume ha f aifie he uual (local) Lipchiz and bounded-growh condiion which guaranee exience and (pahwie) uniquene of a (rong) oluion x } of (2.). Under hee condiion, here exi a coninuou verion of x }. Therefore we may aume ha he pah ω x (ω) are coninuou for P-almo all ω Ω. We inroduce he noaion P,x for he law of he proce x }, aring in x a ime, and ue E,x o denoe expecaion wih repec o P,x. Noe ha he ochaic proce x } i an (inhomogeneou) Markov proce. We are inereed in fir exi ime of x from pace ime e. Le A R [, ] be Borel-meaurable. Auming ha A conain (x, ), we define he fir exi ime of (x, ) from A by τ A = inf [, ]: (x, ) A }, (2.2) and agree o e τ A (ω) = for hoe ω Ω which aify (x (ω), ) A for all [, ]. For convenience, we hall call τ A he fir exi ime of x from A. Typically, we will conider e of he form A = (x, ) R [, ]: g () < x < g 2 ()} wih coninuou 5

6 funcion g < g 2. Noe ha in hi cae, τ A i a opping ime 2 wih repec o he canonical filraion of (Ω, F, P) generaed by x }. Before urning o he precie aemen of our reul, le u inroduce ome noaion. We hall ue y for y o denoe he malle ineger which i greaer han or equal o y, and y z and y z o denoe he maximum or minimum, repecively, of wo real number y and z. By g(u) = O(u) we indicae ha here exi δ > and K > uch ha g(u) Ku for all u [, δ], where δ and K of coure do no depend on or σ. Similarly, g(u) = O(u) i o be underood a lim u g(u)/u =. From ime o ime, we wrie g(u) = OT () a a horhand for lim T up u T g(u) =. Finally, le u poin ou ha mo eimae hold for mall enough only, and ofen only for P-almo all ω Ω. We will re hee fac only when confuion migh arie. 2.2 Nonbifurcaing equilibria We ar by conidering he nonlinear SDE (2.) in he cae of f admiing a nonbifurcaing equilibrium branch. We will make he following aumpion. Aumpion 2.. There exi an inerval I = [, T ] or [, ) and a conan d > uch ha he following properie hold: here exi a funcion x : I R, called equilibrium branch, uch ha f(x (), ) = I; (2.3) f i wice coninuouly differeniable wih repec o x and for x x () d and I, wih uniformly bounded derivaive. In paricular, here exi a conan M > uch ha xx f(x, ) 2M in ha domain; he linearizaion of f a x (), defined a a() = x f(x (), ), (2.4) i bounded away from zero, ha i, here exi a conan a > uch ha a() a I. (2.5) We need no addiional aumpion on σ in hi ecion. However, he reul are only of inere for σ = O(). In he deerminiic cae σ =, he following reul i known (ee Fig. a): Theorem 2.2 (Deerminiic cae [Ti, Gr]). Conider he equaion dx d = f(x, ). (2.6) There are conan, c, c >, depending only on f, uch ha for <, (2.6) admi a paricular oluion x de uch ha x de x () c I; (2.7) 2 For a general Borel-meaurable e A, he fir exi ime τ A i ill a opping ime wih repec o he canonical filraion, compleed by he null e. 6

7 if x x () c and a() a for all I (ha i, when x () i a able equilibrium branch), hen he oluion x de of (2.6) wih iniial condiion x de = x aifie x de x de x x de e a /2 I. (2.8) Remark 2.3. The paricular oluion x de i ofen called low oluion or adiabaic oluion of Equaion (2.6). I i no unique in general, a uggeed by (2.8). We reurn now o he SDE (2.) wih σ >. Le u fir conider he able cae, ha i, we aume ha a() a < for all I. We aume ha a =, x ar a ome (deerminiic) x ufficienly cloe o x (). Theorem 2.2 how ha he deerminiic oluion x de of x () exponenially fa. wih he ame iniial condiion x de = x reache a neighbourhood of order We are inereed in he ochaic proce y = x x de due o noie from he deerminiic oluion x de dy = [ f(x de, which decribe he deviaion. I obey he SDE + y, ) f(x de, ) ] d + σ dw, y =. (2.9) We will prove ha y remain in a neighbourhood of wih high probabiliy. I i inrucive o conider fir he linearizaion of (2.9) around y =, which ha he form where dy = ā()y d + σ dw, (2.) ā() = x f(x de, ) = a() + O() + O ( x x () e a /2 ). (2.) Taking and x x () ufficienly mall, we may aume ha ā() i negaive and bounded away from zero. The oluion of (2.) wih arbirary iniial condiion y i given by y = y e α()/ + σ e α(,)/ dw, α(, ) = ā(u) du, (2.2) where we wrie α(, ) = α() for breviy. Noe ha α(, ) con( ) whenever. If y ha variance v, hen y ha variance v() = v e 2α()/ + σ2 e 2α(,)/ d. (2.3) Since he fir erm decreae exponenially fa, he iniial variance v i forgoen a oon a e 2α()/ i mall enough, which happen already for > O( log ). For y =, (2.2) implie in paricular ha for any δ >, P, y δ } e δ2 /2v(), (2.4) and hu he probabiliy of finding y, a any given I, ouide a rip of widh much larger han 2v() i very mall. Our fir main reul ae ha he whole pah x } of he oluion of he nonlinear equaion (2.) lie wih high probabiliy in a imilar rip, cenred around x de. We only need o make one conceion: he widh of he rip ha o be bounded away from zero. Therefore, we define he rip a B (h) = (x, ) R I : x x de < h ζ() }, (2.5) 7

8 where ζ() = 2 ā() e2α()/ + e 2α(,)/ d. (2.6) Here σ 2 ζ() can be inerpreed a he variance (2.3) a ime of he proce (2.2) aring wih iniial variance v = σ 2 /(2 ā() ). We hall how in Lemma 3. ha ζ() = 2 a() + O() + O( x x () e a /2 ). (2.7) Le τ B(h) denoe he fir exi ime of x from B (h). Theorem 2.4 (Sable cae). There exi, d and h, depending only on f, uch ha for <, h h and x x () d, where P,x τ B(h) < } C(, ) exp 2 h 2 σ 2 [ O() O(h) ] }, (2.8) C(, ) = α() 2 + 2, (2.9) and O() and O(h) do no depend on. The proof, given in Secion 3., i divided ino wo main ep. Fir, we how ha an eimae of he form (2.8), bu wihou he erm O(h), hold for he oluion of he linearized equaion (2.). Then we how ha whenever y < h ζ() for, he bound y < h( + O(h)) ζ() almo urely hold for. Remark 2.5. The reul of he preceding heorem remain rue when /2 ā() in he definiion (2.6) of ζ() i replaced be an arbirary ζ, provided ζ >. The erm O( ) may hen depend on ζ. Noe ha ζ() and v()/σ 2 are boh oluion of he ame differenial equaion z = 2ā()z +, wih poibly differen iniial condiion. If x x () = O(), ζ() i an adiabaic oluion (in he ene of Theorem 2.2) of he differenial equaion, aying cloe o he equilibrium branch z = / 2ā(). The eimae (2.8) ha been deigned for iuaion where σ, and i ueful for σ h. We expec he exponen o be opimal in hi cae, bu did no aemp o opimize he prefacor C(, ), which lead o ubexponenial correcion. If we aume, for inance, ha σ = q, q >, and ake h = p wih < p < q, (2.8) can be wrien a P,x τ B(h) < } ( + 2 ) exp [ O() O( p 2 2(q p) ) O( 2(q p) log ) ]}. (2.2) The -dependence of he prefacor i o be expeced. I i due o he fac ha a ime increae, he probabiliy of x ecaping from a neighbourhood of x de alo increae, bu very lowly if σ i mall. The eimae (2.8) how ha for a fracion γ of rajecorie o leave he rip B (h), we have o wai a lea for a ime γ given by α( γ ) = γ 2 h 2 [ ] } exp O() O(h) 2 σ 2 2 2, (2.2) which i compaible wih reul on he auonomou cae. Le u now conider he unable cae, ha i, we now aume ha he linearizaion a() = x f(x (), ) aifie a() a > for all I. Theorem 2.2 how he exience of 8

9 a paricular oluion x de of he deerminiic equaion (2.6) uch ha x de x () c for all I. We define ā() = x f( x de, ) = a() + O() > and α() = ā() d. The linearizaion of (2.) around x de again admi a oluion y of he form (2.2). Unlike in he able cae, he variance (2.3) grow exponenially fa (a lea wih e a/ ), and hu one expec he probabiliy of x remaining cloe o x de o be mall. Indeed, if ρ y, hen we have P,y up y < ρ } P,y y < ρ } = ρ y e α()/ 2πv() dx ρ y eα()/ e x2/2v() 2ρ 2πv(), (2.22) which goe o zero a ρσ e α()/ for. In hi eimae, however, we neglec all rajecorie which leave he inerval ( ρ, ρ) before ime and come back. We will derive a more precie eimae for he general, nonlinear cae. Thi i he conen of he econd main reul of hi ecion. Le } B u (h) = (x, ) R I : x x de h < (2.23) 2ā() and denoe by τ Bu(h) he fir exi ime of x from B u (h). Theorem 2.6 (Unable cae). There exi and h, depending only on f, uch ha for all h σ h, all and all x aifying (x, ) B u (h), we have P,x τ Bu(h) } e exp κ σ2 α() } h 2, (2.24) where κ = π 2e( O(h) O() ) doe no depend on. The proof, given in Secion 3.2, i baed on a pariion of he inerval [, ] ino mall inerval, and a comparion of he nonlinear equaion wih i linearizaion on each inerval. The above reul how ha x i unlikely o remain in B u (h) a oon a h 2 /σ 2. A major limiaion of (2.24) i ha i require h σ. Obaining an eimae for larger h i poible, bu require coniderably more work. We will provide uch an eimae in he more difficul, bu alo more inereing cae of he pichfork bifurcaion, ee Theorem 2. below. 2.3 Pichfork bifurcaion We now conider he SDE (2.) in he cae of f undergoing a pichfork bifurcaion. We will aume he following. Aumpion 2.7. There exi a neighbourhood N of he origin (, ) uch ha f i hree ime coninuouly differeniable wih repec o x and in N, and here exi a conan M > uch ha xxx f(x, ) 6M for all (x, ) N ; f(x, ) = f( x, ) for all (x, ) N ; f exhibi a upercriical pichfork bifurcaion a he origin, i.e., x f(, ) =, x f(, ) > and xxx f(, ) <. (2.25) 9

10 By recaling x and, we may and will aume ha x f(, ) =, x f(, ) = and xxx f(, ) = 6 (2.26) a in he andard cae f(x, ) = x x 3. Noe ha he aumpion ha f be odd i no neceary for he exience of a pichfork bifurcaion. However, he deerminiic yem behave very differenly if x = i no alway an equilibrium. The mo naural iuaion in which f(, ) = for all i he one where f i odd. The proof below can eaily be exended o he cae where f i no necearily odd provided f exhibi a upercriical pichfork bifurcaion wih x() being he equilibrium branch which become unable a he bifurcaion poin. Uing (2.26), we find ha he linearizaion of f a x = aifie a() = x f(, ) = + O( 2 ). (2.27) A andard reul of bifurcaion heory [GH, IJ] ae ha under hee aumpion, here exi a neighbourhood N N of (, ) in which he only oluion of f(x, ) = are he line x = and he curve x = ±x (), x () = [ + O() ],. (2.28) If N i mall enough, he equilibrium x = i able for < and unable for >, while x = ±x () are able equilibria wih linearizaion a () = x f(x (), ) = 2 [ + O() ]. (2.29) The only oluion of x f(x, ) = in N are he curve x = ± x(), x() = /3 [ + O() ],. (2.3) If f i four ime coninuouly differeniable, he erm O() in he la hree equaion can be replaced by O(). We briefly ae wha i known for he deerminiic equaion dx d = f(x, ), (2.3) where we ake an iniial condiion (x, ) N wih x > and <, ee Fig. b. Oberve ha α(, ) = a() d i decreaing for < < and increaing for >. Definiion 2.8. The bifurcaion delay i defined a Π( ) = inf > : α(, ) > }, (2.32) wih he convenion Π( ) = if α(, ) < for all >, for which α(, ) i defined. One eaily how ha Π( ) i differeniable for ufficienly cloe o, and aifie lim Π( ) = and lim Π ( ) =. Theorem 2.9 (Deerminiic cae). Le x de be he oluion of (2.3) wih iniial condiion = x. Then here exi conan, c, c depending only on f, and ime x de = + O( log ) 2 = Π( ) = Π( ) O( log ) 3 = Π( ) + O( log ) (2.33)

11 uch ha, if < x c, < and (x de, ) N, < x de c e α(, )/ for 2 x de x () c for 3. (2.34) The proof i a raighforward conequence of differenial inequaliie, ee for inance [Ber, Propoiion 4.6 and 4.8]. We now conider he SDE (2.) for σ >. The reul in hi ecion are only inereing for σ = O( ), while one of hem (Theorem 2.) require a condiion of he form σ log σ 3/2 = O( ) (where we have no ried o opimize he exponen 3/2). Le u fix an iniial condiion (x, ) N wih <. For any T (, ), we can apply Theorem 2.4 on he inerval [, T ] o how ha x T i likely o be a mo of order σ δ + c e α( T, )/ for any δ >. We can alo apply he heorem for > T o how ha he curve ±x () arac nearby rajecorie. Hence here i no limiaion in conidering he SDE (2.) in a domain of he form x d, T where d and T can be aken mall (independenly of and σ of coure!), wih an iniial condiion x T = x aifying x d. From now on, we will alway aume ha N = (x, ) R 2 : x d, T } (2.35) for ome d, T >. We fir how ha x i likely o remain mall for T. Acually, i urn ou o be convenien o how ha x remain cloe o he oluion x e α(, T )/ of he linearizaion of (2.3). We define he variance-like funcion ζ() = 2 a( T ) e2α(, T )/ + e 2α(,)/ d. (2.36) T We hall how in Lemma 4.2 ha for ufficienly mall, here exi conan c ± uch ha c ζ() c + for T, (2.37) c ζ() c+ for. (2.38) The funcion ζ() i ued o define he rip B(h) = (x, ) [ d, d] [ T, ]: x x e α(, T )/ < h ζ() }. (2.39) Le τ B(h) denoe he fir exi ime of x from B(h). Theorem 2. (Behaviour for ). There exi conan and h, depending only on f, T and d, uch ha for <, h h, x h/ /4 and T, where P T,x τ B(h) < } C(, ) exp 2 C(, ) = h 2 σ 2 [ r() O ( )]} h 2 (2.4) α(, T ) + O() 2 (2.4) and r() = O() for T, and r() = O( ) for.

12 The proof (given in Secion 4.2) and he inerpreaion of hi reul are very cloe in piri o hoe of Theorem 2.4. The only difference lie in he kind of -dependence of he error erm. The eimae (2.4) i ueful when σ h, and how ha he ypical preading of pah around he deerminiic oluion will lowly grow unil =, where i i of order σ/ /4, ee Fig. 2. Le u now examine wha happen for. We fir how ha x i likely o leave quie oon a uiably defined region D conaining he line x =. We define D = D(κ) by D(κ) = (x, ) [ d, d] [, T ]: x f(x, ) κa() }, (2.42) where κ (, ) i a free parameer. The upper and lower boundary of D(κ) are given by ± x(), where x() aifie x() = κ OT () a(). (2.43) Laer, we will aume κ (/2, 2/3). Thi will implify he analyi of he dynamic afer x ha lef D(κ). The upper bound on κ implie x() x() x (), while he lower bound guaranee ha he oluion of he correponding deerminiic equaion doe no reurn o D(κ) once i ha lef hi e, cf. Propoiion 4. below. Le τ D(κ) denoe he fir exi ime of x from D(κ). Then we have he following reul. Theorem 2. (Ecape from D(κ)). Chooe κ (, 2/3) and le (x, ) D(κ). Aume ha σ log σ 3/2 = O( ). Then for T, P,x τd(κ) } C x() ( log σ a() + α(, ) ) e κα(, )/ σ, (2.44) e 2κα(, )/ where C > i a (numerical) conan. The proof of hi reul (given in Secion 4.3) i by far he mo involved of he preen work. We ar by eimaing, in a imilar way a in Theorem 2.6, he fir exi ime from a rip S of widh lighly larger han σ/ a(). The probabiliy of reurning o zero afer leaving S can be eimaed; i i mall bu no exponenially mall. However, he probabiliy of neiher leaving D(κ) nor reurning o zero i exponenially mall. Thi fac can be ued o devie an ieraive cheme ha lead o he exponenial eimae (2.44). We poin ou ha for any ube D D(κ), we have he rivial eimae P,x τ D } P,x τ D(κ) }, and hu (2.44) alo provide an upper bound for he fir exi ime from maller e. Le u finally conider wha happen afer he pah ha lef D(κ) a ime τ = τ D(κ) (wih κ (/2, 2/3)). Fir noe ha (2.43) immediaely implie ha for T and x x(), x f(x, ) ã() = x f( x(), ) ηa() provided η 2 3κ OT (). (2.45) Le x de,τ denoe he oluion of he deerminiic equaion (2.3) aring in x() a ime τ (he cae where one ar a x() i obained by ymmery). We hall how in Propoiion 4. ha x de,τ alway remain beween x() and x (), and approache x () according o x de,τ ( ) = x () O 3/2 O ( τ e ηα(,τ)/). (2.46) 2

13 Moreover, deerminiic oluion aring a differen ime approach each oher like x de, x de,τ ( x de, τ x(τ) ) e ηα(,τ)/ [τ, T ]. (2.47) The linearizaion of f a x de,τ aifie a τ () = x f(x de,τ, ) = a () + O For given τ, we conruc a rip A τ (h) around x de,τ of he form ( ) + O ( e ηα(,τ)/). (2.48) A τ (h) = (x, ): τ T, x x de,τ < h ζ τ () }, (2.49) where he funcion ζ τ () i defined by ζ τ () = and aifie 2 ã(τ) e2ατ (,τ)/ + ζ τ () = τ e 2ατ (,)/ d, α τ (, ) = a τ (u) du, (2.5) ( ) 2 a () + O 3 + O( e ηα(,τ)/), (2.5) cf. Lemma 4.2. Le τ A τ (h) denoe he fir exi ime of x from A τ (h). Theorem 2.2 (Approach o x ). Le κ (/2, 2/3). Then here exi conan and h, depending only on f, T and d, uch ha for <, h < h τ and τ T, where P τ, x(τ) τ A τ (h) < } C τ (, ) exp 2 h 2 [ σ 2 ( h )]} O() O τ (2.52) C τ (, ) = ατ (, τ) a () d + 2. (2.53) The proof i given in Secion 4.4. Thi reul i ueful for σ h τ, and how ha he ypical preading of pah around x de,τ i of order σ/, ee Fig Dicuion Le u now examine ome of he conequence of hee reul. Fir of all, hey allow o characerize he influence of addiive noie on he bifurcaion delay. In he deerminiic cae, hi delay i defined a he fir exi ime from a rip of widh around x =, ee Theorem 2.9. A poible definiion of he delay in he ochaic cae i hu he fir exi ime τ delay from a imilar rip. An appropriae choice for he widh of he rip i x( ) = O( /4 ), ince uch a rip will conain B(h) for every admiible h, and he par of he rip wih will be conained in D(κ). Theorem 2. and 2. hen imply ha if, P T,x τ delay < } C(, ) e O(/σ2 ) P T,x τ delay } C x() a() log σ σ ( + α(, ) (2.54) ) e κα(, )/ e. (2.55) 2κα(, )/ If we chooe in uch a way ha α(, ) = c log σ for ome c >, he la expreion reduce o P T,x τ delay } = O ( σ κc log σ 2), (2.56) 3

14 which become mall a oon a c > /κ. The bifurcaion delay will hu lie wih overwhelming probabiliy in he inerval [, O ( log σ )]. (2.57) Theorem 2.2 implie ha for ime larger han O( log σ ), he pah are unlikely o reurn o zero in a ime of order. The wilde behaviour of he pah i o be expeced in he inerval (2.57), becaue a region of inabiliy i croed, where x f >. Our reul on he pichfork bifurcaion require σ, while he eimae (2.57) i ueful a long a σ i no exponenially mall. We can hu diinguih hree regime, depending on he noie ineniy: σ : A modificaion of Theorem 2. how ha for < σ, he ypical preading of pah i of order σ/. Near he bifurcaion poin, he proce i dominaed by noie, becaue he drif erm f x 3 i oo weak o counerac he diffuion. Depending on he global rucure of f, an appreciable fracion of he pah migh ecape quie early from a neighbourhood of he bifurcaion poin. In ha iuaion, he noion of bifurcaion delay become meaningle. e /p σ for ome p < : The bifurcaion delay lie in he inerval (2.57) wih high probabiliy, where log σ ( p)/2 i ill microcopic. σ e K/ for ome K > : The noie i o mall ha he pah remain concenraed around he deerminiic oluion for a ime inerval of order. The ypical preading i of order σ ζ(), which behave like σ e α()/ / /4 for, ee Lemma 4.2. Thu he pah remain cloe o he origin unil α() log σ K. If log σ > α(π( )) = α( ), hey follow he deerminiic oluion which make a quick raniion o x () a = Π( ). The expreion (2.57) characerizing he delay i in accordance wih experimenal reul [TM, SMC], and wih he approximae calculaion of he la croing of zero [JL]. The numerical reul in [Ga], which are fied, a =., o τ delay σ.5 for weak noie and τ delay e 85 σ for rong noie, eem raher myeriou. Finally, he reul in [Ku], who approximae he probabiliy deniy by a Gauian cenered a he deerminiic oluion, can obviouly only apply o he regime of exponenially mall noie. Noe ha he eimae (2.57) ugge how o chooe he peed a which he bifurcaion parameer i wep when deermining a bifurcaion diagram experimenally: Since we wan he bifurcaion delay o be microcopic, hould no exceed a cerain value depending on he noie ineniy σ. In fac, repeaing he experimen for differen value of yield an eimae for log σ. On he oher hand, increaing arificially he noie level σ allow o work wih larger weeping rae, reducing he ime co of he experimen. Anoher inereing queion i how fa he pah concenrae near he equilibrium branche ±x (). The deerminiic oluion, aring a x( ) a ome ime >, all rack x () a a diance which i aympoically of order / 3/2. Therefore, we can chooe one of hem, ay x de,, and meaure he diance of x from ha deerminiic oluion. We reric our aenion o hoe pah which are ill in a neighbourhood of he origin a ime, a mo pah are. We wan o how ha for uiably choen (, ) and (, ), mo pah will leave D(κ) unil ime and reach a δ-neighbourhood of x de, 4

15 a ime τ D(κ) +. Le u eimae P,x ( τ D(κ) <, P },x τd(κ) + E,x up [τ D(κ) +,] τd(κ) < } P τ D(κ), x(τ D(κ) ) ) c } x x de, < δ up [τ D(κ) +,] x x de, }} δ. (2.58) The fir erm decreae roughly like σ e κα(, )/ and become mall a oon a α(, ) log σ. The econd ummand i bounded above by con E,x τd(κ) < } exp 2 [ σ 2 δ O ( τd(κ) e ηα(τ D(κ)+,τ D(κ) )/ ) ]}} 2. (2.59) Therefore, δ hould be large compared o σ/ and we alo need ha i a lea of order O( log σ ). Then we ee ha afer a ime of order O( log σ ), he ypical pah will have lef D(κ) and, afer anoher ime of he ame order, will reach a neighbourhood of, which cale wih σ/. Finally, we can alo eimae he probabiliy of reaching he poiive raher han he negaive branch. Conider x, aring in x a ime <, and le >. Wihou lo of generaliy, we may aume ha x >. The ymmery of f implie x de, P,x x } = 2 P,x [, ) : x = }, (2.6) and herefore i i ufficien o eimae he probabiliy for x o reach zero before ime zero, for inance. We linearize he SDE (2.) and ue he fac ha he oluion x of he linearized equaion dx = a()x d + σ dw, x = x (2.6) aifie x x a long a x doe no reach zero. For he Gauian proce x we know P,x [, ) : x = } ( = 2 P,x x }) = u() e y2 /2 dy, (2.62) 2π u() where u() = x e α(,)/ / v(, ) and v(, ) denoe he variance of x. For =, u() i of order x /4 σ e con 2 /, ee Lemma 4.2. Thu he probabiliy in (2.62) i exponenially cloe o one for mall, and we conclude ha he probabiliy for x o reach he poiive branch raher han he negaive one i exponenially cloe o /2. When he global behaviour of f i known, we can alo inveigae he long-ime behaviour of he oluion x. For inance, in he pecial cae f(x, ) = x x 3, under he aumpion σ 2 con/ log, i i unlikely ha a pah which i cloe o one of he able equilibrium branche ± a ome ime of order, will wich o he oher equilibrium branch again. Thi i a conequence of he fac ha he diance beween he equilibrium branche increae while he branche become more and more aracive. Along he line of Secion 3. i can be hown ha he probabiliy of ever reaching zero again decay like e con/σ2 in ha cae. 5

16 3 The moion near nonbifurcaing equilibria In hi ecion we conider he nonlinear SDE (2.) under Aumpion 2. which guaranee he exience of a hyperbolic equilibrium branch. Secion 3. i devoed o he able cae, while in Secion 3.2, we conider he unable cae. 3. Sable cae We fir conider he cae of a able equilibrium, ha i, we aume ha a() a for all I. We will ar by analying he linearizaion of (2.) around a given deerminiic oluion. Propoiion 3.4 how ha he oluion of he linearized equaion are likely o remain in a rip of widh h ζ() around he deerminiic oluion. Here ζ() i relaed o he variance and will be analyed in Lemma 3.. Propoiion 3.7 allow o compare he rajecorie of he linear and he nonlinear equaion, and hu complee he proof of Theorem 2.4. By Theorem 2.2, here exi a c > uch ha he deerminiic oluion x de of (2.6) wih iniial condiion x de = x aifie x de x () 2c + x x () e a /2 I, (3.) provided x x () c. Le x denoe he oluion of he SDE (2.), aring a ime = in ome x. We are inereed in he ochaic proce y = x x de, which decribe he deviaion due o noie from he deerminiic oluion x de. I obey an SDE of he form dy = [ā()y + b(y, ) ] d + σ dw, y =, (3.2) where we have inroduced he noaion ā() = ā () = x f(x de, ) b(y, ) = b (y, ) = f(x de + y, ) f(x de, ) ā()y. (3.3) Taking and x x () ufficienly mall, we may aume ha here exi a conan d > uch ha x de + y x () d whenever y d. I follow from Taylor formula ha for all (y, ) [ d, d] I, b(y, ) My 2 (3.4) ā() a() M ( 2c + x x () e a /2 ). (3.5) We may furher aume ha here are conan ā + ā > a /4 uch ha Finally, he relaion ā () = x f(x de a conan c 2 > uch ha ā + ā() ā I. (3.6), ) + xx f(x de, ) f(xde, ) implie he exience of ( ā () c 2 + x x () e a /2 ). (3.7) Our analyi will be baed on a comparion beween oluion of (3.2) and hoe of he linearized equaion dy = ā()y d + σ dw, y =. (3.8) 6

17 I oluion y a ime i a Gauian random variable wih mean zero and variance v() = σ2 e 2α(,)/ d where α(, ) = ā(u) du. (3.9) Noe ha (3.6) implie ha α(, ) ā ( ) whenever, which implie in paricular, ha v() i no larger han σ 2 /2ā. We can, however, derive a more precie bound, which i ueful when and e a/2 are mall. To do o, we inroduce he funcion ζ() = 2 ā() e2α()/ + e 2α(,)/ d, where α() = α(, ). (3.) Noe ha v() σ 2 ζ(), and ha boh funcion differ by a erm which become negligible a oon a > O( log ). The behaviour of ζ() i characerized in he following lemma. Lemma 3.. The funcion ζ() aifie he following relaion for all I. ζ() = 2 ā() + O() + O( x x () e a /2 ) (3.) ζ() 2ā + 2ā (3.2) ζ () Proof: By inegraion by par, we obain ha ζ() = Uing (3.6) and (3.7) we ge ā () ā() 2 e2α(,)/ d c 2 ā 2 c 2 2ā 3 + c 2 ā 2 2ā() 2 e 2ā ( )/ d + c 2 x x () ā 2 (3.3) ā () ā() 2 e2α(,)/ d. (3.4) e [ 2ā ( ) a /2]/ d x x () 2ā a /2 e a /2, (3.5) which prove (3.). We now oberve ha ζ() i a oluion of he linear ODE dζ d = ( ) 2ā()ζ +, ζ() = 2 ā(). (3.6) Since ζ() > and ā() <, we have ζ () /. We alo ee ha ζ () whenever ζ() /2ā + and ζ () whenever ζ() /2ā. Since ζ() belong o he inerval [/2ā +, /2ā ], ζ() mu remain in hi inerval for all. A we have already een in (2.4), he probabiliy of finding y ouide a rip of widh much larger han 2v() i very mall. By Lemma 3., we now know ha 2v() behave approximaely like σ a() /2. One of he key poin of he preen work i o how ha he whole pah y } remain in a rip of imilar widh wih high probabiliy. The rip will be defined wih he help of ζ() inead of v(), becaue we need he widh o be bounded away from zero, even for mall. 7

18 To inveigae y we need o eimae he ochaic inegral eα(,u)/ dw u. To do o, we would like o ue he Bernein-ype inequaliy } δ 2 } P up ϕ(u) dw u δ exp 2, (3.7) ϕ(u)2 du valid for Borel-meaurable deerminiic funcion ϕ(u). Unforunaely, hi eimae canno be applied direcly, becaue in our cae, he inegrand depend explicily on he upper inegraion limi. Thi i why we inroduce a pariion of he inerval [, ]. Lemma 3.2. Le ρ : I R + be a meaurable, ricly poiive funcion. Fix K N, and le = u u < < u K = be a pariion of he inerval [, ]. Then P, up σ ρ() } e α(,u)/ dw u h 2 where P k = exp h 2 ( )( 2 σ 2 inf ρ() 2 e 2α(u k,)/ u k u k Proof: We have P, up σ } e α(,u)/ dw u h ρ() = P, k,..., K} : up u k u k ρ() 2 K k= P, up u k u k uk e α(u)/ dw u h σ K P k, (3.8) k= ) } e 2α(uk,)/ d. (3.9) e α(,u)/ h } dw u σ inf ρ() e α()/}. u k u k Applying he Bernein inequaliy (3.7) o he la expreion, we obain (3.8). (3.2) Remark 3.3. Noe ha in he proof of Lemma 3.2 we have no ued he monooniciy of α(, ) o ha he eimae (3.8) can alo be applied in he cae where ā() change ign. We are now ready o derive an upper bound for he probabiliy ha y leave a rip of appropriae widh hρ() before ime. Taking ρ() = ζ() will be a good choice ince i lead o approximaely conan P k in (3.8). Propoiion 3.4. There exi an r = r(ā +, ā ) uch ha P, y } up h C(, ) exp ζ() 2 h 2 } σ 2 ( r), (3.2) where C(, ) = α() (3.22) 8

19 Proof: Le α() K = 2 2. (3.23) For k =,..., K, we define he pariion ime u k by he relaion α(u k ) = 2 2 k, (3.24) which i poible ince α() i coninuou and decreaing. Thi definiion implie in paricular ha α(u k, u k ) = 2 2 and, herefore, u k u k 2 2 /ā. Bounding he inegral in (3.9) by ζ(u k ), we obain P k exp 2 We have e 2α(u k,)/ e 4 and h 2 σ 2 ζ() ζ(u k ) = Since ζ(u k ) /2ā +, hi implie P k exp 2 and he reul follow from Lemma 3.2. inf u k u k uk ζ() ζ(u k ) e2α(u k,)/ }. (3.25) ζ (u) du u k. (3.26) h 2 ( ) σ 2 4ā+ e 4}, (3.27) ā Remark 3.5. If we only aume ha ā i Borel-meaurable wih ā() ā for all I, we ill have P, up y h/ } 2ā C(, ) exp h 2 2 σ 2 e 4}. (3.28) To prove hi, we chooe he ame pariion a before and bound he inegral in (3.9) by /2ā. We now reurn o he nonlinear equaion (3.2), he oluion of which we wan o compare o hoe of i linearizaion (3.8). To hi end, we inroduce he even Ω (h) = ω : y (ω) < h } ζ() [, ] (3.29) Ω (h) = ω : y(ω) < h } ζ() [, ]. (3.3) Propoiion 3.4 give u an upper bound on he probabiliy of he complemen of Ω (h). The key poin o conrol he nonlinear cae i a relaion beween he e Ω and Ω (for lighly differen value of h). Thi i done in Propoiion 3.7 below. Noaion 3.6. For wo even Ω and Ω 2, we wrie Ω a.. Ω 2 if P-almo all ω Ω belong o Ω 2. Propoiion 3.7. Le γ = 2 2ā + M/ā 2 and aume ha h < d ā /2 γ. Then Ω (h) a.. ( [ Ω γ ) + 4 h] h (3.3) Ω (h) a.. Ω ( [ + γh ] h ). (3.32) 9

20 Proof:. The difference z = y y aifie wih z = P-a.. Now, which implie dz d = [ā()z + b(y + z, ) ] (3.33) z = z e α(,u)/ b(y u + z u, u) du, (3.34) for all [, ]. 2. Le u aume ha ω Ω (h). Then we have for all [, ] and hu by (3.35), e α(,u)/ b(y u, u) du (3.35) y (ω) h ζ() h 2ā d 2, (3.36) z (ω) α(,u)/ Mh2 e du. (3.37) 2ā The inegral on he righ-hand ide can be eimaed by (3.2), yielding Therefore, e α(,u)/ du 2ζ 2 () ā. (3.38) z (ω) Mh2 2ā 2 M ā + h 2ā 2 h ζ(), (3.39) which prove (3.3) becaue y (ω) y (ω) + z (ω). 3. Le u now aume ha ω Ω (h). Then we have y (ω) d/2 for all [, ] a in (3.36). For δ = γh, we have δ < by aumpion. We conider he fir exi ime and he even τ = inf [, ]: z δh ζ() } [, ] } (3.4) A = Ω (h) ω : τ(ω) < }. (3.4) If ω A, hen for all [, τ(ω)], we have y (ω) ( + δ)h ζ() d, and hu by (3.35) and (3.38), z (ω) e α(,u)/ M( + δ)2 h 2 2ā du M( + δ)2 h 2 2ā 2 < δh ζ(). (3.42) However, by he definiion of τ, we have z τ(ω) (ω) = δh ζ(τ(ω)), which conradic (3.42) for = τ(ω). Therefore PA} =, which implie ha for almo all ω Ω, we have z (ω) < δh ζ() for all [, ], and hence for hee ω, which prove (3.32). y (ω) < ( + δ)h ζ() [, ] (3.43) 2

21 We cloe hi ubecion wih a corollary which i Theorem 2.4, reaed in erm of he proce y. I i a direc conequence of Propoiion 3.7 and 3.4. Corollary 3.8. There exi h and, depending only on f, uch ha for < and h < h, P, y } up > h C(, ) exp h 2 [ ] } O() O(h) ζ() 2 σ 2. (3.44) 3.2 Unable cae We now conider a imilar iuaion a in Secion 3., bu wih an unable equilibrium branch, ha i, we aume ha a() a > for all I. Our aim i o prove Theorem 2.6 which i equivalen o Propoiion 3. below. The proof i again baed on a comparion of oluion of he nonlinear equaion (2.) and i linearizaion around a given deerminiic oluion. Theorem 2.2 how he exience of a paricular oluion x de of he deerminiic equaion (2.6) uch ha x de x () c for all I. We are inereed in he ochaic proce y = x x de, which decribe he deviaion due o noie from hi deerminiic oluion x de. I obey he SDE where dy = [ā()y + b(y, ) ] d + σ dw, (3.45) ā() = ā () = x f( x de, ) b(y, ) = b (y, ) = f( x de + y, ) f( x de, ) ā()y (3.46) are he analog of ā and b defined in (3.3). Taking ufficienly mall, we may aume ha here exi conan ā, ā, d >, uch ha he following eimae hold for all I and all y uch ha y d: ā() ā, ā () ā, b(y, ) My 2. (3.47) The bound on ā () i a conequence of he analog of (3.7) ogeher wih he fac ha x de x () = O(). We fir conider he linear equaion dy = ā()y d + σ dw. (3.48) Given he iniial value y, he oluion y mean y eα()/ and variance a ime i a Gauian random variable wih v() = σ2 e 2α(,)/ d, (3.49) where α(, ) = ā(u) du ā ( ) for. The variance, which i growing exponenially fa, can be eimaed wih he help of he following lemma. Lemma 3.9. For < < 2ā 2 /ā, one ha e 2α(,)/ d = [ e 2α()/ 2ā() ] [ ] + O(). (3.5) 2ā() 2

22 Proof: By inegraion by par, we obain ha e 2α(,)/ d = 2ā() e2α()/ 2ā() 2 ā () ā() 2 e2α(,)/ d, (3.5) which implie ha [ 2 ā ā 2 ] e 2α(,)/ d 2ā() e2α()/ [ 2ā() + 2 ā ā 2 ] e 2α(,)/ d. (3.52) By our hypohei on, he fir erm in bracke i poiive. The following propoiion, which reae Theorem 2.6 in erm of y, i he main reul of hi ubecion. Propoiion 3.. There exi conan, h > uch ha for all h σ h, all and for any given y wih y 2ā() < h, we have where κ = π 2e( O(h) O() ). P,y up y } 2ā() < h e exp κ σ2 α() } h 2, (3.53) Proof:. Le K N and le = u < u < < u K = be any pariion of he inerval [, ]. We define he even A k = ω : up y } 2ā() < h u k u k+ B k = ω : y uk } (3.54) 2ā(u k ) < h A k. Le q k be a deerminiic upper bound on P k = P u k,y uk A k }, valid on B k. Then we have by he Markov propery P,y y } 2ā() < h up K } = P,y A k = E,y } } K 2 k= A E,y AK y k } uk = E,y k= } K 2 k= A P K k K 2 q K P,y k= A k } K k= q k. (3.55) 2. To define he pariion, we e K = γ α() σ 2 h 2 (3.56) for ome γ (, ] o be choen laer, and α(u k+, u k ) = γ h2, k =,..., K 2. (3.57) σ2 22

23 Since α(u k+, u k ) ā (u k+ u k ), we have u k+ u k h2 γ σ 2 ā, and uing Taylor formula, we find for all [u k, u k+ ] and all k =,..., K h2 ā σ 2 γ ā() h2 ā + ā(u k ) σ 2 γ, (3.58) ā 2 where ā i he upper bound on ā, ee (3.47). In order o eimae P k, we inroduce linear approximaion (y (k) ) [uk,u k+ ] for k,..., K 2}, defined by dy (k) = ā()y(k) ā 2 + σ dw (k), y (k) u k = y uk, (3.59) where W (k) = W W uk i a Brownian moion wih W u (k) k = which i independen of W : u k }. If ω A k, we have for all [u k, u k+ ] y (ω) y (k) (ω) Mh2 2ā u k e α(,u)/ b(y u, u) du e α(uk+,uk)/ ā(u k ) where r = M e(2ā 3 ) /2 + O(). Thi how ha on A k, [ + O() ] r h 2 2ā(), (3.6) y (k) (ω) [ + r h ] h 2ā() [u k, u k+ ]. (3.6) 3. We are now ready o eimae P k. (3.6) how ha on B k, P k P u k,y uk up } 2ā() < h( + r h) u k u k+ y (k) P u k,y uk y (k) u k+ 2ā(u k+ ) < h( + r h) } (3.62) 2h( + r h) 2πv u (k) 2ā(uk+ ), k+ where v u (k) k+ denoe he condiional variance of y u (k) k+, given y uk. A in (3.5), v u (k) k+ = σ2 uk+ [ e 2α(uk+,)/ d = σ2 e 2α(u k+,u k ] )/ [ ] + O(). (3.63) u k 2 ā(u k ) ā(u k+ ) I follow ha ā(u k+ )v u (k) k+ σ2 2 σ2 2 [ e 2γh2 /σ 2 ā(u k+) ā(u k ) [( + 2γ h2 σ 2 )( ā γh 2[ ā 2ā 2 γh 2[ O() ]. ] [ ] O() h 2 ) ā 2 σ 2 γ ( ) ] [ ] + 2γ O() ] [ ] O() Inering hi ino (3.62), we obain for each k =,..., K 2 on B k he eimae (3.64) P k 2h( + r h) [ ] [ ] + O() = + O() + O(h) =: q. (3.65) 2π 2γh 2 πγ 23

24 Noe ha for any γ (/π, ], here exi h > and > uch ha q < for all h h and all. Since q K = i an obviou bound, we obain from (3.55) P,y up y } 2ā() < h q K q exp α() σ 2 h 2 2γq 2 q2 log ( /q 2)}. (3.66) Chooing γ o ha q 2 = / e hold, yield almo he opimal exponen, and we obain P,y up y } 2ā() < h e exp κ α() σ 2 } h 2. (3.67) 4 Pichfork bifurcaion 4. Preliminarie In hi ecion, we conider he nonlinear SDE (2.) in he cae where f undergoe a upercriical pichfork bifurcaion, i.e., we require Aumpion 2.7 o hold in a region N = (x, ) R 2 : x d, T } N. The noie ineniy σ i aumed o aify σ = O( ) hroughou Secion 4. Only in Subecion 4.3, hi condiion will be lighly renghened o σ log σ 3/2 = O( ). Recall ha we recaled pace and ime in order o obain (2.26). Uing Taylor erie and he ymmery aumpion, we may wrie for all (x, ) N f(x, ) = a()x + b(x, ) = x [ a() + g (x, ) ] x f(x, ) = a() + g (x, ) where a(), g (x, ), g (x, ) are wice coninuouly differeniable funcion aifying a() = x f(, ) = + O( 2 ) (4.) g (x, ) = [ + γ (x, ) ] x 2 g (x, ) Mx 2 (4.2) g (x, ) = [ 3 + γ (x, ) ] x 2 g (x, ) 3Mx 2, wih γ, γ ome coninuou funcion uch ha γ (, ) = γ (, ) =. The following andard reul from bifurcaion heory i eaily obained by applying he implici funcion heorem, ee [GH, p. 5] or [IJ, Secion II.4] for inance. We ae i wihou proof. Propoiion 4.. If T and d are ufficienly mall, here exi wice coninuouly differeniable funcion x, x : (, T ] R + of he form x () = [ + OT () ] wih he following properie: x() = /3 [ + OT () ] (4.3) he only oluion of f(x, ) = in N are eiher of he form (, ), or of he form (±x (), ) wih > ; he only oluion of x f(x, ) = in N are of he form (± x(), ) wih ; he derivaive of f a ±x () i a () = x f(x (), ) = 2 [ + OT () ]. (4.4) 24

25 he derivaive of x () and x() aify dx d = 2 [ + O T ()], d x d = 2 3 [ + O T ()]. (4.5) A already poined ou in Secion 2.3, here i no rericion in auming T and d o be mall. Thu we may aume ha he erm OT () are ufficienly mall o do no harm. For inance, we may and will alway aume ha a () <. In he following ubecion, we are going o analye he dynamic in hree differen region of he (, x)-plane: near x = for, near x = for, and near x = x () for. Fir, in Subecion 4.2, we analye he behaviour for. Theorem 2. i proved in he ame way a Theorem 2.4, he main difference lying in he behaviour of he variance which i inveigaed in Lemma 4.2. Subecion 4.3 i devoed o he raher involved proof of Theorem 2.. We ar by giving ome preparaory reul. Propoiion 4.7 eimae he probabiliy of remaining in a maller rip S in a imilar way a Propoiion 3.. We hen how in Lemma 4.8 ha he pah are likely o leave D(κ) a well, unle he oluion of a uiably choen linear SDE reurn o zero. The probabiliy of uch a reurn o zero i udied in Lemma 4.9. Finally, Theorem 2. i proved, he proof being baed on an ieraive cheme. The la ubecion analye he moion afer τ D(κ). Here, he main difficuly i o conrol he behaviour of he deerminiic oluion, which are hown o approach x (), cf. Propoiion 4.. We hen prove ha he pah of he random proce are likely o ay in a neighbourhood of he deerminiic oluion. The proof i imilar o he correponding proof in Secion The behaviour for We begin by conidering he linear SDE wih iniial condiion x = x a ime [ T, ). Le dx = a()x d + σ dw (4.6) v(, ) = σ2 e 2α(,)/ d (4.7) denoe he variance of x. A before, we now inroduce a funcion ζ() which will allow u o define a rip ha he proce x i unlikely o leave before ime, ee Corollary 4.5 below. Le ζ() = 2 a( ) e2α(, )/ + e 2α(,)/ d. (4.8) The following lemma decribe he behaviour of ζ(). Lemma 4.2. There exi conan c ± > uch ha c ζ() c + for c ζ() c+ for (4.9) c e 2α()/ ζ() c+ e 2α()/ for T. If, moreover, a () > on [, ], hen ζ() i increaing on [, ]. 25

26 Proof: Fir noe ha Equaion (4.2) implie he exience of conan a + a > uch ha a + a() a for T a a() a + for T. (4.) For, hi implie a + ( 2 2 ) 2α(, ) a ( 2 2 ). Inegraion by par yield he relaion e a ±(2 2 )/ d = 2a ± e a±(2 2 )/ 2a ± e a±(2 2)/ 2a ± 2 d. (4.) Since he la wo erm on he righ-hand ide are negaive, he upper bound for i immediae. For he correponding lower bound, we ue ζ() e2α(, )/ 2 a( ) 2 e a +( 2 2 )/ d e a +(( 2) 2 2 )/, (4.2) 2a + 2 where he la inequaliy i obained by replacing e a +( 2 2 )/ by on he righ-hand ide of (4.). For /2, we hu ge ζ() /(4a + ), while for /2 <, we find ζ() ( e 3a + )/(4a + ). In he cae, we ue he relaion ζ() = ζ( ) e 2α(, )/ + e 2α(,)/ d. (4.3) Since α(, ) = O() for,, we conclude ha ζ() remain of order / for. For, we have e 2α()/ ζ() = ζ( ) e 2α( )/ + e 2α()/ d. (4.4) Now, 2α() a 2 for implie ha he righ-hand ide remain of order / for all. Finally, aume ha a () > for all, and recall ha ζ() i he oluion of he iniial value problem dζ d = 2a() ζ +, ζ( ) = 2 a( ). (4.5) Since ζ(), ζ > for all poiive. For negaive, ζ i poiive whenever he funcion V () = ζ() + /2a() i negaive. We have V ( ) = and dv d = 2a() V a () 2a() 2. (4.6) Since V < whenever V =, V can never become poiive. Thi implie ζ. The following propoiion how ha he oluion x of he linearized equaion (4.6) i likely o rack he oluion of he correponding deerminiic equaion. 26

27 Propoiion 4.3. Aume ha T <. For ufficienly mall, P,x up x x e α(, )/ ζ() } > h C(, ) exp 2 h 2 σ 2 [ r() ] }, (4.7) where C(, ) = α(, ) 2 + a and where r() = O() for, and r() = O( ) for. (4.8) Proof: We will only give he proof in he cae a hi i he more inereing par. By Lemma 3.2, he probabiliy in (4.7) i bounded by 2 K k= P k, where P k = exp 2 h 2 σ 2 ζ(u k ) } inf ζ(u) e 2α(u k,u)/ (4.9) u k u u k for any pariion = u < < u K = of he inerval [, ]. The choice of he pariion hould reflec he differen behaviour of x for and for. We e α(, ) + K = 2 2, K = K + (4.2) and define he pariion ime by α(u k, ) = 2 2 k for k K, u k = + (k K ) for K k K. (4.2) Eimaing P k a in he proof of Propoiion 3.4, we obain P k exp P k exp Finally, le u noe ha h 2 ( 2 σ 2 2 ) e 4} for k K, (4.22) a c h 2 ( ) } 2 σ 2 [ + 2a + c + ] e a + for K k K. (4.23) c 2K α(, ) ( + ) + 4 α(, ) 2 + a , (4.24) which conclude he proof of he propoiion. Le u now compare oluion of he wo SDE dx = a()x d + σ dw x = x (4.25) dx = f(x, ) d + σ dw x = x, (4.26) where [ T, ). We define he even Ω (h) = ω : x (ω) x e α(, )/ h } ζ() [, ] (4.27) Ω (h) = ω : x (ω) x e α(, )/ h } ζ() [, ]. (4.28) Propoiion 4.3 give u an upper bound on he probabiliy of he complemen of Ω (h). We now give relaion beween hee even. 27