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EXISTENCE OF A UNIQUE STRONG SOLUTION TO THE DMPK EQUATION MAXIMILIANBUTZ ABSTRACT. Forthetransmissionofelectronsinaweaklydisorderedstripofmaterial Dorokhov, Mello, Pereyra and Kumar (DMPK) proposed a diffusion process for the transfer matrices. The correspoding transmission eigenvalues satisfy the DMPK stochasticdifferentialequations,likeDysonBrownianmotioninthecontextofGOE/GUE randommatrices. WecontrolthesingularrepulsiontermsofthisSDEwithastopping- timeargument,anditsdegenerateinitialconditionbyanapproximationprocedure,and therebyestablishtheDMPKequationtobewellposed. 3 1 0 2 n 1. INTRODUCTION a J F.J.Dyson[11]introducedanN ×N matrix-valuedBrownianmotion,tobeprecise 8 astochasticprocess(X2(t))t≥0, ] R (1) X (t) = Y (t)+Y (t)∗, 2 2 2 P withY (t)beinganN×N matrixwithallentriesindependentcomplexstandardBrow- . 2 h nianmotions. t a X (t)takesvaluesinthecomplexhermitianmatrices,andisdistributedlikeascaled 2 m GUEmatrixforfixedtimet. UsingtheItoˆ formulaandtheinvarianceoftheincrements [ under unitary transformations, it is possible to prove that the law in path space of 2 the eigenvalues λ1(t),...,λN(t) of X2(t) is the unique weak solution of the system of v stochasticdifferentialequations(SDE) 6 9 2 (cid:88) 1 (2) dλ (t) = √ dW (t)+2 dt, k = 1,...,N 3 k β k λ (t)−λ (t) k l 3 l(cid:54)=k . 5 with Wk independent standard Brownian motions and β = 2 in our case. Although it 0 is not hard to formally derive (2), the singular level repulsion terms (which originate 2 from second order perturbation theory) cause that a solution of (2) is only defined up 1 : to the first collision of two eigenvalues. Thus, to make the derivation of (2) from (1) v rigorous, one needs to prove that almost surely no level crossing can occur. This was i X accomplisheddirectlybyanalysisof(1)in[17],andin[7],[21]viaaLyapunovfunction r argumentfor(2)(forgeneralβ ≥ 1),whichispresentedcomprehensivelyin[2]. a Among other applications in physics, random matrices have been employed to un- derstandthepropertiesofdisorderedconductors,inparticularthephenomenonofuni- versal conductance fluctuations [1, 14]. In the arguably simplest case, one can model the conductor as a quasi-1D wire, a system of N interacting channels. The matrix in considerationisthetransfermatrixM ∈ C2N×2N whichmapsthequantumstateatthe Date:January9,2013. 1 2 M.BUTZ left side of the wire to the state at the right side (rather than mapping incoming states tooutgoingstates,asascatteringmatrixdoes). Instead of the self-adjointness of Wigner matrices, the main constraint for transfer matrices is current conservation, (corresponding to unitarity of the scattering matrix), whichinasuitablebasisreads (cid:18) (cid:19) 1 0 (3) M∗Σ M = Σ , Σ = N , z z z 0 −1 N [19],andwedefinethegroup (4) G = (cid:8)M ∈ C2N×2N : M∗Σ M = Σ (cid:9). 2 z z Incasetheunderlyingquantummechanicalsystemistime-reversalsymmetric,i.e. inthe absenceofmagneticfields,thetransfermatrixadditionallyhastosatisfy (cid:18) (cid:19) 0 1 (5) Σ MΣ = M, Σ = N . x x x 1 0 N Accordingly, (6) G = (cid:8)M ∈ C2N×2N : M∗Σ M = Σ andΣ MΣ = M(cid:9). 1 z z x x InadditiontothegroupsG withparameterβ = 1,2asdefinedabove,thevalueβ = 4 β isalsoalgebraicallyandphysicallymeaningful,asitcorrespondstoaquantumsystem allowing for spin-orbit scattering, [5]. However, this last case does not introduce any new stochastic features and is disregarded for the sake of notational simplicity. One should remark that the correspondence of symmetries of the quantum system and the choiceofmatrixensemblesalreadyarisesforGaussianensemblesandDysonBrownian motion [18], the ensemble suitable for time-reversal invariant quantum systems (and thusloselyrelatedtoourG )being 1 (7) X (t) = Y (t)+Y (t)T, 1 1 1 withY (t)beinganN ×N matrixwithallentriesindependentrealstandardBrownian 1 motions. As in the theory of Wigner matrices, the focus is on the spectral properties of those transfermatrices: foreachMthereexistsadecomposition (cid:18)M M (cid:19) (cid:18)U 0 (cid:19)(cid:32) Λ (cid:0)Λ2−1(cid:1)21(cid:33)(cid:18)V 0 (cid:19) (8) M = ++ +− = + + M−+ M−− 0 U− (cid:0)Λ2−1(cid:1)12 Λ 0 V− with (non-unique) N × N unitary matrices U , U or V , V , and a diagonal matrix + − + − Λ ≥ 1 . ForgeneralM ∈ G ,thereisnorelationshipbetweenthoseunitaries,whilefor N 2 M ∈ G ,U = U andV = V . ForlaterusewedefinethesubmanifoldsG˜ astheset 1 + − + − β ofthosematricesinG withadecompositionsuchthatΛ > 1 andhasnondegenerate β N eigenvalues. Forphysicallyrealisticconductors,Mwillbearandomquantityduetomicroscopic disorder, so that on a macroscopic level it is natural to model the transfer matrix as a stochasticprocess(M(s)) dependingonthelengthsofthewire. s≥0 DMPKPROCESS 3 If we combine two pieces of wire with transfer matrices M and M , respectively, I II weexpecttheconcatenatedwiretohave (9) M = M M II I as its transfer matrix [19] and if we furthermore assume that the transfer matrices of disjointshortpiecesofwirearei.i.d.,wearriveatthestochasticdifferentialequation (10) dM(s) = dL(s)M(s) (s ≥ 0). with a suitable stochastic process (L(s)) that encodes the scattering properties of s≥0 shortwirepiecesandischosensuchthat(3)(and,forβ = 1,(5))areconserved. Stochas- tically, every intial distribution for (10) makes sense, but it is physically reasonable to assigntoazero-lengthconductorthetransfermatrix (11) M(0) = 12N. TheprocessLwewillactuallyconsiderisa2N ×2N matrix-valuedBrownianmotion (cid:18) a (s) b(s) (cid:19) + (12) L(s) = b∗(s) a (s) − withN ×N blockprocessesa ,a ,bwhich,intermsof(1,7),aredistributedlike + − √ d d a = a = iX / 2N + − 2 √ d d (13) Reb = Imb = X /(2 N +1) (β = 1) 1 √ d b = Y / N (β = 2). 2 Forβ = 1,RebandImbaremutuallyindependentandjointlyindependentofa ,while ± a and a are correlated by a = a . For β = 2, however, the processes a , a and b + − − + + − areallindependent. One motivation for this choice of L is that it predicts the same dynamics for the diagonal matrix Λ as the maximum entropy assumption [19]. The reason is, that after endowing the groups G with a suitable right-invariant metric, (10) is just the Brown- β ian motion on the Riemannian manifold G , while [19] essentially single out the heat β kernel by their maximum entropy requirement for the propagator. Moreover, (10) has beenderivedin[3,4]asscalinglimitfromamicroscopicquantummodel(anAnderson modelonatube)—uptominordeviationsoriginatingfromtheAndersonHamiltonian dictatingapreferredbasisintheweak-perturbationlimit. Finally,andmostimportantinthepresentcontext,theincrementsofLareinvariant undertheconjugation (14) U∗dLU =d dL. U canbeanyunitarymatrixoftheform (cid:18) (cid:19) U 0 (15) U = + 0 U − 4 M.BUTZ withtheblocksU andU chosenindependentlyforβ = 2,andcorrelatedbyU = U + − + − in case β = 1, compare (21), (22) in [3]. This perfectly corresponds to the orthogo- nal/unitaryinvarianceoftheGUE/GOE-likeincrementsof(1,7),andwillfinallyallow ustoproveanautonomous(i.e. eigenvector-independent)equationforΛ. First, we give a formal derivation and start with a stochastic evolution for the matrix M∗ M and its eigenvalues λ , ordered like λ ≥ ... ≥ λ . Inserting the ++ ++ ++ k 1 N componentof(10), (16) dM = daM +dbM , ++ ++ −+ intoItoˆ’sformula,weobtain d(M∗ M ) =M∗ (da M +dbM ) ++ ++ ++ + ++ −+ +(cid:0)M∗ da∗ +M∗ db∗(cid:1)M ++ + −+ ++ (17) +(cid:0)M∗ da∗ +M∗ db∗(cid:1)(da M +dbM ) ++ + −+ + ++ −+ =M∗ dbM +M∗ db∗M +M∗ M ds+M∗ M ds ++ −+ −+ ++ ++ ++ −+ −+ where we used the independence of a and b, and their explicit form (13) in the last + line. By second order perturbation theory for the eigenvalues of M∗ M and Itoˆ’s ++ ++ formula, (18) dλ = (cid:10)v ,d(cid:0)M∗ M (cid:1)v (cid:11)+(cid:88) (cid:12)(cid:12)(cid:10)vl,d(cid:0)M∗++M++(cid:1)vk(cid:11)(cid:12)(cid:12)2, k k ++ ++ k λ −λ k l l(cid:54)=k withv the eigenvectors ofM∗ M . In view of (8), v are the columns ofV∗, while √ k ++ ++ k + λ = Λ arethesingularvaluesofM . Furthermore,by k kk ++ U∗M V∗ = (cid:0)Λ2−1(cid:1)12 (19) − −+ + U∗M V∗ = Λ + ++ + itisconvenienttoconsidertheprocesswith (20) db˜(s) = U∗(s)db(s)U (s) + − forwhich,by(14), (21) db˜ =d db. Withthesedefinitions, dλ =2(cid:112)λ (λ −1)d(cid:16)Reb˜ (cid:17)+(2λ −1)ds k k k kk k +(cid:88)(cid:16)(cid:112)λ (λ −1)db˜ +(cid:112)λ (λ −1)db˜ (cid:17) (22) l k lk k l kl l(cid:54)=k ×(cid:16)(cid:112)λ (λ −1)db˜ +(cid:112)λ (λ −1)db˜ (cid:17)/(λ −λ ). l k lk k l kl k l Wecanuse(21)with(13)tosee (cid:115) 1 (23) Reb˜ (s) = − B (s) kk k β(N −1)+2 DMPKPROCESS 5 forallk = 1,...,N,withindependentstandardrealBrownianmotionsB ,while k (cid:16)(cid:112)λ (λ −1)db˜ +(cid:112)λ (λ −1)db˜ (cid:17)(cid:16)(cid:112)λ (λ −1)db˜ +(cid:112)λ (λ −1)db˜ (cid:17) l k lk k l kl l k lk k l kl (24) β = (2λ λ −λ −λ )ds, k l k l β(N −1)+2 andthus   (cid:115) β (cid:88) 2λkλl −λk −λl 4λk(λk −1) (25) dλk = 2λk −1+ ds− dBk. β(N −1)+2 λ −λ β(N −1)+2 k l l(cid:54)=k After introducing the transmission eigenvalues T = λ−1 ∈ [0,1], another application of k k the Itoˆ formula yields the DMPK (Dorokhov-Mello-Pereyra-Kumar) equation for the transmissioneigenvalues(compare(3.9)in[5],(24)in[3]): (26) dT (s) = v (T(s))ds+D (T(s))dB (s), k k k k B ,k = 1,...,N independentBrownianmotions, k   2Tk β (cid:88) Tk +Tj −2TkTj vk = −Tk + 1−Tk +  βN +2−β 2 T −T k j j(cid:54)=k (27) (cid:115) T2(1−T ) D = 4 k k . k βN +2−β Remark. As already noted in [3], the term “DMPK equation” usually refers to the for- wardequationforthedensityoftheT ’s. Wealsouseitforthecorrespondingstochastic k differentialequation,asthisisthemorenaturalobjectinouranalysis. The form of both the drift term v and the diffusion term D causes the transmission k k valuestodecayto0asstendstoinfinity,whichperfectlymatchesthedecreaseofcon- ductance with increasing wire length. However, the drift v also contains repulsion k termsoriginatingfromsecondorderperturbationtheory. Asaconsequence,theeigen- values T “try to avoid” degeneracy. Thus, and similar to Dyson Brownian motion, k theabovederivationismerely“formal”,astheItoˆ formulaisonlyapplicableifthede- nominator λ − λ (or T − T ) stays away from zero, i.e. M∗ (s)M (s) never has k l k l ++ ++ degenerate eigenvalues. For s > 0, this is already a nontrivial finding, but even more, wewanttostarttheevolutionequation(10)withthecompletelydegeneratematrix(11). The goal of our paper is the derivation of the following rigorous statement about the transmissionvalueprocessof(M(s)) . s≥0 Theorem 1. Let β = 1 or β = 2, and (M(s)) be the solution of (10), starting from (11). s≥0 Thenthedistributioninpathspaceofitstransmissioneigenvalueprocessisgivenbythelawof theuniquecontinuousprocess(T (s)) whichstartsfromT (0) = 1fork = 1,...,N k s≥0,k=1...N k and is a strong solution to (26) for s > 0. Moreover, the transmission eigenvalues T , T for k l k (cid:54)= lalmostsurelynevercollidefors > 0,witheachstayingintheopeninterval(0,1). Speaking in terms of manifolds, we thus see that the paths of the Brownian motion (10) on G , even though starting from a degenerate matrix M(0) ∈ G \G˜ , are almost β β β surely contained in G˜ for all positive s, and do not explode for finite s (this would β 6 M.BUTZ correspond to T (s) = 0). It will be obvious from the proof of Theorem 4 that the k way to handle the singular initial condition (11) could as well be applied to any other degenerate M(0) ∈ G \ G˜ . For the sake of notational simplicity, we stay with the β β physicallymostinterestingcaseM(0) = 1 . 2N WewillstartfromconsideringtheDMPKequationnotasaneigenvalueprocess,but aprocessinitsownrightandshowthatithasauniquestrongsolutionforallβ ≥ 1in subsection2.1. Oncethisisestablished,weverifyinsubsection2.2thatthetransmission eigenvalueprocessofMhasthesamelawinpathspace. Wheneverpossible,theanalysiswillbecarriedoutinthepictureofthetransmission eigenvalues T (rather than the λ ). This is because the T are the physically most k k k intuitivequantity,eachT representingthetransmittanceofonechannel. Accordingly, k whenmodellingthepassageofelectronsthroughaN-channelwire,thedimensionless conductanceofthewireisgivenbytheLandauerformula([5],equation(33)), N (cid:88) (28) g = T . k k=1 This formula for g and other linear functionals of the T describe the most important k transport properties of the disordered conductor. Plugging the DMPK forward equa- tioninto(28),[6]establishedtheuniversalconductancefluctuations 2 (29) Var(g) = 15β forβ = 2intheintermediatemetallicphasewith1 (cid:28) s (cid:28) N. 2. RIGOROUS ANALYSIS OF THE PROCESS OF TRANSMISSION EIGENVALUES 2.1. TheuniquestrongsolutionoftheDMPKequation. Wefirstinvestigateequation (26) starting from a nondegenerate initial condition 0 < T (0) < ... < T (0) < 1, 1 N correspondingtoM(0) ∈ G˜ . β Theorem 2. Let (Ω,P) be a probability space endowed with a filtration F = (F ) and t t≥0 B ,...,B be independent real Brownian motions adapted to F. Let D be the set of strictly 1 N N ordered N-tupels in (0,1), D = {x ∈ R : 0 < x < ... < x < 1}. Assume initial condi- N 1 N tions T(0) = (T (0),...,T (0)) ∈ D , and take β ≥ 1. Then the stochastic differential 1 N N equation(26)hasauniquestrongsolutionwithinitialconditionT(0), andT(s) ∈ D forall N s > 0. Thelawofthisstrongsolutionalsodefinestheuniqueweaksolutionof(26). Proof. Followingtheideasof[2]. ForR > 1,weregularizethesingularrepulsioninthe drifttermof(27)bysetting    (R) 2Tk β (cid:88) Tk +Tj −2TkTj vk (T) = χR(T)−Tk + βN +2−β 1−Tk + 2 T −T  (30) k j j(cid:54)=k (R) D (T) = χ (T)D (T) k R k with χ a smooth cut-off function compactly supported in the interior of D and R √ N χ (T) = 1 whenever T ∈ D and dist(T,∂D ) ≥ ( 2R)−1. Now v(R) and D(R) R N N k k are both uniformly Lipschitz, so the modified system has a unique strong solution DMPKPROCESS 7 (cid:0)T(R)(s)(cid:1) andauniqueweaksolution([15],Theorems5.2.5,5.2.9,[20],Lemma5.3.1). s≥0 TheideaoftheproofisthatT(R)(s)shouldbeasolutionoftheoriginalDMPKequation (R) (26) as long as the differences of the T and their distance to the boundaries of [0,1] k arelargerthanR−1. TakingR tozeroshouldthenyieldasolutionto(26)foralltimes. To control the separation of the eigenvalues and their distance to 0 and 1, we define a Lyapunovfunction   N N (cid:88) (cid:88) (31) f(T) = −2log(|Tk|)−2log(|1−Tk|)− log(|Tk −Tl|) k=1 l=1,l(cid:54)=k It will be enough to define f for T from D , as we will always use it together with N stoppingtimeswhichwillmakesureT doesn’tleavethisset. Note that if we take T ∈ D , all the logarithms in (31) are positive, thus we have N f(T) ≥ 0andalso (32) −log(|T −T |) ≤ f(T). k l Define (cid:110) (cid:16) (cid:17) (cid:111) (33) S = inf s ≥ 0 : f T(R)(s) ≥ M M,R for M > 0. As f is smooth on sets where it is uniformly bounded, S is a stopping M,R time. Ontheevent{S > s},wehave M,R (34) −log(|T(R)(s)−T(R)(s)|) ≤ f(T(R)(s)) ≤ M k l soifwetakeM = M(R) = logR, |T(R)(s)−T(R)(s)| ≥ e−M = R−1 k l (35) |T(R)(s)| ≥ e−M = R−1 k |1−T(R)(s)| ≥ e−M = R−1. k Thus, up to S , T(R) and all T(R˜) with R˜ ≥ R coincide and T(R) also solves the M(R),R originalequation(26)uptoS . M(R),R Therefore, to calculate df(s) up to the stopping time S = S it is enough to M M(R),R plug(26)intoItoˆ’sformula([15],Theorem3.3.3). Wedroptheoverscript(R),asitdoes 8 M.BUTZ notmatterinthiscontext. (36) df(T(s)) =   N N (cid:88) 1 1 (cid:88) 1 2 − + −  T 1−T T −T k k k j k=1 j=1,j(cid:54)=k    N 2Tk β (cid:88) Tk +Tl −2TkTl ×−Tk + 1−Tk + ds βN +2−β 2 T −T k l l=1,l(cid:54)=k   +(cid:88)N  1 + 1 + (cid:88)N 1 4 Tk2(1−Tk) ds T2 (1−T )2 (T −T )2 βN +2−β k=1 k k j=1,j(cid:54)=k k j +dA(s), with(A(s)) denotingthelocalmartingalefulfilling s≥0   (cid:115) N N (cid:88) 1 1 (cid:88) 1 1−Tk (37) dA(s) = 2 − + − 2Tk dBk(s). T 1−T T −T βN +2−β k k k j k=1 j=1,j(cid:54)=k (Remark: Because of (35), A(s ∧ S ) is certainly a martingale, which is enough for M our purpose. We only see that A is really a local martingale after having verified that S (cid:37) ∞asM → ∞.) M IntroducingT ≡ 0,T ≡ 1,wecanmoreconvenientlyrewritedf 0 N+1   N N+1 4 (cid:88) (cid:88) 1 df(T(s)) =−   βN +2−β T −T k j k=1 j=0,j(cid:54)=k   N+1 (38) ×(β−1)Tk2+βNTk(Tk −1)+β (cid:88) Tkρ−kTlds l=0,l(cid:54)=k   N N+1 (cid:88) (cid:88) 1 ρk +  4 ds (T −T )2 βN +2−β k j k=1 j=0,j(cid:54)=k +dA(s), DMPKPROCESS 9 wherewehavesetρ = T2(1−T ). Usingρ = ρ = 0,wehave k k k 0 N+1     N N+1 N+1 N+1 (cid:88) (cid:88) 1 (cid:88) ρk (cid:88) ρk Z =   −  T −T T −T (T −T )2 k j k l k m k=1 j=0,j(cid:54)=k l=0,l(cid:54)=k m=0,m(cid:54)=k     N+1 N+1 N+1 N+1 (cid:88) (cid:88) 1 (cid:88) ρk (cid:88) ρk =   −  T −T T −T (T −T )2 k j k l k m k=0 j=0,j(cid:54)=k l=0,l(cid:54)=k m=0,m(cid:54)=k N+1N+1N+1 (cid:88) (cid:88) (cid:88) ρk = 1(l (cid:54)= k,j (cid:54)= k,l (cid:54)= j) (T −T )(T −T ) k l k j (39) k=0 j=0 l=0 N+1N+1N+1 (cid:18) (cid:19) (cid:88) (cid:88) (cid:88) 1 1 1 = 1(l (cid:54)= k,j (cid:54)= k,l (cid:54)= j)ρ − k T −T T −T T −T k j k l j l k=0 j=0 l=0 N+1N+1N+1 (cid:18) (cid:19) (cid:88) (cid:88) (cid:88) ρk −ρj ρk −ρl 1 =−2Z + 1(l (cid:54)= k,j (cid:54)= k,l (cid:54)= j) − T −T T −T T −T k j k l j l k=0 j=0 l=0 N+1N+1N+1 1 (cid:88) (cid:88) (cid:88) = 1(l (cid:54)= k,j (cid:54)= k,l (cid:54)= j)(1−T −T −T ), j l k 3 k=0 j=0 l=0 andthusforT ∈ D N 2 (40) |Z| ≤ (N +1)3. 3 Furthermore (41) N N+1 N+1N+1 (cid:88) (cid:88) Tk(1−Tk) (cid:88) (cid:88) Tk(1−Tk)−Tj(1−Tj) 1 (cid:88) = 1(j (cid:54)= k) = (1−T −T ), k j T −T 2(T −T ) 2 k j k j k=1j=0,j(cid:54)=k k=0 k=0 k(cid:54)=j so (cid:12) (cid:12) (cid:12) N N+1 (cid:12) (cid:12)(cid:88) (cid:88) Tk(1−Tk)(cid:12) (N +1)(N +2) (42) (cid:12) (cid:12) ≤ . (cid:12) T −T (cid:12) 2 k j (cid:12)k=1j=0,j(cid:54)=k (cid:12) forallT ∈ D . Finally, N (cid:88)N N(cid:88)+1 T2 (cid:88)N (cid:88)N T2 (cid:88)N T2 k = − T + k + k k T −T 1−T T −T j k k j k k=1j=0,j(cid:54)=k k=1 k=1 j,k=1,j(cid:54)=k (43) = −(cid:88)N (cid:88)N Tk +Tj +(cid:88)N Tk2 2 1−T k k=1j=1 k=1 andthusonD N (cid:88)N N(cid:88)+1 T2 (cid:88)N T2 (44) −N2 ≤ k ≤ k . T −T 1−T j k k k=1j=0,j(cid:54)=k k=1 10 M.BUTZ Thus there is a continous bounded function g with g(T) ≤ CN2 for T ∈ D such N that df(T(s))    = g(T(s))ds+ 4(β−1) (cid:88)N  Tk2 − N(cid:88)+1 Tk2(1−Tk)ds+dA(s) βN +2−β 1−T (T −T )2 (45) k=1 k j=0,j(cid:54)=k k j   = g(T(s))ds− 4(β−1) (cid:88)N  (cid:88)N Tk2(1−Tk)ds+dA(s). βN +2−β (T −T )2 k j k=1 j=0,j(cid:54)=k Recallingthatβ ≥ 1andA(s∧S )isamartingale, M (46) E(f(T(s∧S ))) ≤ CN2s+f(T(0)). M Asf isnonnegative,wehavetheMarkovestimate 1 CN2s+f(T(0)) (47) P(S ≤ s) ≤ E(f(T(s∧S ))) ≤ M M M M foralls ≥ 0andM > 0. Nowasbydefinition (48) SM˜ = SM(R˜),R˜ ≥ SM(R),R = SM forR˜ ≥ R,weconcludefromP(S ≤ s) → 0asM → ∞foralls ≥ 0that M (49) S (cid:37) ∞ (M → ∞) M almost surely. As we have already seen that T(R) is a solution to (26) up to S , M(R),R settingT = T(R) uptoS forallR > 1providesuswithastrongsolutionof(26) M(R),R fors ∈ [0,∞). Thisfinishesthestrongexistencepart. To prove strong uniqueness, let a strong solution T of (26) be given and define the stoppingtimeS˜ quiteanalogouslytoS ,butthistimebasedonT insteadofT(R), M M (50) S˜ = inf{s ≥ 0 : f(T(s)) ≥ M}. M IfwetakeeM = R,wehavearesultlike(35),andbystronguniquenessofT(R),T and T(R) coincideuptoS˜ . Asonecanshow M (51) S˜ (cid:37) ∞ (M → ∞) M just as before, T has to be the solution we obtained in the existence part almost surely foralltimes. WeakuniquenessofT followsbyexactlythesamereasoningfromtheweakunique- nesspropertyofT(R) forallR > 1. Finally, T(s) ∈ D for all s ≥ 0 almost surely, because we have seen that for any N s ≥ 0 there is almost surely a (random) R > 1 such that T(t) = T(R)(t) for all t ≤ s, but the T(R) stay in D almost surely due to the cutoff in the definitions of v(R) and N k D(R). (cid:3) k