On a Fractional Stochastic Landau-Ginzburg Equation

Transcription

1 Applied Mahemaical Sciences, Vol. 4, 1, no. 7, On a Fracional Sochasic Landau-Ginzburg Equaion Nguyen Tien Dung Deparmen of Mahemaics, FPT Universiy 15B Pham Hung Sree, Hanoi, Vienam dungn@fp.edu.vn Tran Hung Thao Insiue of Mahemaics Vienamese Academy of Science and Technology No 18 Hoang-Quoc-Vie Road, Hanoi, Vienam hhao@mah.ac.vn Absrac The aim of his paper is o invesigae a fracional sochasic Landau- Ginzburg equaion for modelling superconduciviy from an approximaion approach by he fac ha a fracional Brownian moion of Liouville form can be approximaed by semimaringales in L -space.the exisence and uniqueness of he soluion are proved and is explici form is found as well. Mahemaics Subjec Classificaion: 8D35, 8B3, 6H3 Keywords: Landau-Ginzburg heory, fracional Brownian moion, semimaringale 1 Inroducion I is well known ha he Landau-Ginzburg heory is a mahemaical heory used o model superconduciviy [4]. This heory examines he macroscopic properies of a superconducor wih he aid of general hermodynamic argumens. Landau and Ginzburg esablished ha he free energy of a superconducor near he semiconducing ransiion can be expressed in erms of an order parameer, his parameer describes how deep ino he superconducing phase he sysem is. The module X of his parameer depending on ime can

2 318 N. T. Dung and T. H. Thao be considered as he sysem sae saisfying he sochasic Landau-Ginzburg equaion of he form dx =( X 3 + bx )d + σx dw, (1.1.1) where W is a sandard Brownian moion, b and σ are some consans. One observes ha he model (1.1.1) does no exacly reflec he superconducing sae of he sysem because he fac ha he sae X a each ime can have a long-ime influence upon he sysem while he soluion of (1.1.1) is only a Markov process ha is of no-memory. So one can inroduce a new model as follows: dx =( X 3 + bx )d + σx dw H, (1.1.) where he fracional Brownian moion W H wih he Hurs index H, ( <H< 1) is a cenered Gaussian process of covariance funcion R(s, ) given by R(s, ) = 1 (H + s H s H ). I is known also he he process W H can be represened in he form W H = 1 Γ(H + 1 ) [ U + where Γ sands for he Gamma funcion, and U = ( s) H 1 dws ], ( ( s) α ( s) α) dw s is a process wih absoluely coninuous pahs. One noes ha he long-range dependence propery focuses a he erm B := ( s) H 1 dw s ha is called he fracional Brownian moion of Liouville form (see [5,, 1]). And in his paper we consider he fracional sochasic Landau-Ginzburg equaion given by dx =( X 3 + bx )d + σx db, (1.1.3) where B = ( s) α dw s,α= H 1 σ and b = α +. The soluion whose exisence is assured as shown laer expresses a longmemory sae of he superconduciviy of he sysem.

3 On a fracional sochasic Landau-Ginzburg equaion 319 An approximaion mehod Our mehod is based on a resul on approximaion of he fracional process B = ( s) α dw s by semimaringales given in [1, 6] ha we recall below: For every ε>, we define: Then we have B ε = ( s + ε) α dw s, α = H 1 ( 1, 1 ). (..1) Theorem.1. I. The process {B ε, } is a semimaringale wih following decomposiion where ϕ ε () = B ε = α ( s + ε) α 1 dw s. ϕ ε (s)ds + ε α W, (..) II. The process B ε converges o B in L (Ω) when ε ends. This convergence is uniform wih respec o [,T]. Proof. Refer o [6]. Nex, le us consider he following fracional differenial equaion in a complee probabiliy space (Ω, F, P) dx = ( ) X 3 + bx d + σx db (..3) X = = X, [,T] The soluion of (..3) is a sochasic process such ha X = X + ( ) X 3 s + bx s ds + σ X s db s (..4) where he fracional sochasic inegral of X s db s will be defined as he L -limi X s db ε when ε, if i exiss. The iniial value X is a measurable random variable independen of {B : T }. Our approximaion approach o solving (..3) can be describes as follows:

4 3 N. T. Dung and T. H. Thao For every ε> we invesigae a corresponding approximaion equaion o (..3) dx ε = ( ) (X ε ) 3 + bx ε d + σx ε db ε (..5) X ε = = X where B ε is defined as in (..). Suppose now ha here exiss a soluion Xε of (..5), hen he fac B ε B implies ha he soluion of equaion (..3) will be limi in L (Ω) of he soluion of (..5) when ε. Indeed, from (..3) and (..5) we ge X ε X b(x ε ) b(x ) ds + σ X ε s db ε s X s db ε s + σ XdBs ε X s db s where b(x) = x 3 + bx. Now by using a localized mehod i is enough o consider X ε N and X N a.s. for some N, and hen here exiss a posiive consan L N such ha b(x ε ) b(x ) L N X ε X. Therefore, he inequaliy (a + b + c) 3(a + b + c ) and an Iô inegraion lead us o he following esimaion E X ε X 3L N E Xs ε X s ds +3σ +3σ E XdB ε s =3(L N + σ ε α ) X ε s X s d[b ε ] s X s db s E X ε s X s ds + c(, ε), where c(, ε) =3σ E XdBs ε X s db s asε by definiion of he fracional sochasic inegral. I follows from he laes inequaliy and from an applicaion of he Gronwall s lemma ha E X ε X c(, ε) e 3(L N +σ ε α ). As a consequence, X ε X in L (Ω) when ε ends o zero.

5 On a fracional sochasic Landau-Ginzburg equaion 31 3 Main Resuls From (..) we can rewrie he equaions (..5) as follows dx ε = ( (X ε)3 + bx ε + σα ) ϕε ()X ε d + σε α X ε dw (3.3.1) X ε = = X The sochasic process X ε ϕ ε is no bounded. However, he exisence of he soluion can be proved as in Theorem 3.1 below and we can esablish he uniqueness of he soluion of equaion (..5) or (3.3.1) by inroducing he sequence of sopping imes τ M = inf{ [,T]: (ϕ ε s) ds > M} T, and considering he sequence of corresponding sopped equaions dx ε τ M = ( ) (X τ ε M ) 3 + bx τ ε M + σα ϕ ε ()X τ ε M d + σε α X τ ε M dw. (3.3.) We can verify he coefficiens of (3.3.) saisfy he local Lipschiz condiion. Hence, he uniqueness of he soluion is assured (see, for insance, [3]). Theorem 3.1. The soluion of equaion (..5) can be explicily given by Proof. Pu X ε = e (b 1 σ ε α )+σb ε ( X + According o he Iô formula we have: Y = e σεα W. e ((b 1 σ ε α )s+σb ε s ) ) 1 ds. (1 ) dy = Y σ ε α d σε α dw. (3.3.3) We consider Z = X ε Y and hen an applicaion of he inegraion-by-par formula gives us dz = X ε dy + Y dx ε σ ε α X ε Y d { = e σεα W (Z ) 3 + ( b + σα ϕ ε () 1 σ ε α) } Z d. (3.3.4)

6 3 N. T. Dung and T. H. Thao This is an ordinary Bernouilli equaion of he form: and he soluion Z is given by Z = P ()Z 3 + Q()Z R ( Q(u)du Z = e Z P (s)e where P () = e σεα W,Q() =b 1 σ ε α + σα ϕ ε () and Q(u)du =(b 1 σ ε α ) + σα I(),I() = sr ) 1 Q(u)du ds ϕ ε (s)ds. Hence, he soluion Z of equaion (3.3.4) can be expressed as ( Z = e (b 1 σ ε α )+σα I() Z + ( ) ) 1 e (b 1 σ ε α )s+σα I(s)+σε α W (s) ds. Combining he laes expression and B ε = αi()+εα W we obain he soluion of he approximaion equaion (..5): X ε = e (b 1 σ ε α )+σb ε The proof is hus complee. ( X + e ((b 1 σ ε α )s+σb ε s ) ) 1 ds. Theorem 3.. Suppose ha X is a random variable such ha X > a.s and E[X] <. If H> 1 hen he sochasic process X defined by ( X = e b+σb X + ) 1 e (bs+σbs) ds (3.3.5) is he limi in L (Ω) of X ε. This limi is uniform wih respec o [,T]. Proof. Pu θ ε () =e (b 1 σ ε α )+σb ε and θ() = e b+σb hen i is clear ha for each m 1 here exiss a finie consan M m > such ha E[θ m ε ()] M m, E[θ m ()] M m for every [,T]. Indeed, E[θ m ()] = e mb E[e mσb ]=e mb+ 1 4H m σ H <, and E[θ m ε ()] = e m(b 1 σ ε α )+ 1 4H m σ [(+ε) H ε H] <.

7 On a fracional sochasic Landau-Ginzburg equaion 33 Moreover, applying he Hölder inequaliy we have following esimaes for any m, k 1: E[θ m ε () θk ()] (E θ ε () m ) 1 (E θ() k ) 1 So here exiss a finie consan M m,k > such ha E[θ m ε () θ k ()] M m,k [,T]. (3.3.6) We now can prove ha θ ε () L θ() uniformly wih respec o [,T], i.e: Indeed, we see ha lim sup ε T θ ε () θ() =. (3.3.7) θ ε () θ() θ() 4 exp ( 1 σ ε α + σ(b ε B ) ) 1 4 Using he relaion e x 1=x + o(x), we obain M 4 exp ( 1 σ ε α + σ(b ε B ) ) 1 4 (3.3.8) exp ( 1 σ ε α +σ(b ε B ) ) σ ε α +σ(b ε B ) 4 + o(...) 4 and hus (3.3.7) follows from Corollary.. 1 σ ε α T + σ(b ε B ) 4 + o(...) 4 (3.3.9) We have also ha θ L ε (s)ds θ (s)ds uniformly wih respec o [, T]. Indeed, we have he following esimae: E θ ε(s)ds θ (s)ds E θ ε() θ () ds [,T]. (3.3.1) Once again, an applicaion of he Hölder inequaliy yields for every [,T] E θ ε() θ () = E[ θ ε () θ() A ε ()] θ ε () θ() A ε () (3.3.11) where A ε () = θ ε () θ() ( θ ε ()+θ() ). Using inequaliies of he form (3.3.6) we see ha here exiss a finie consan M 3 > such ha A ε () M 3 [,T]. (3.3.1)

8 34 N. T. Dung and T. H. Thao I follows from (3.3.1),(3.3.11) and (3.3.1) ha E θ ε(s)ds θ (s)ds M 3 T sup T M 3 sup θ ε () θ() T θ ε () θ() [,T]. (3.3.13) The laes inequaliy assures ha sup θε(s)ds θ (s)ds asε. T Pu η ε () =X + θε (s)ds and η() =X + θ (s)ds. From resuls above we can see ha η ε () L η() uniformly wih respec o [,T]. Nex we will show ha η 1 ε () L η 1 () uniformly wih respec o [,T]. (3.3.14) Indeed, we have η ε () X and η() X a.s for every [,T]. The heorem of finie incremens applied o he funcion g(x) =x 1 yields η 1 ε () η 1 () 1 X3 (η ε() η()). By an argumen analogous o he previous one, we ge η 1 ε () η 1 () M η ε () η() [,T]. where M> is a finie consan. And (3.3.14) follows from his esimae. As a consequence we have following asserion X ε = θ ε()η 1 ε () L θ()η 1 () =X (). The proof of heorem is hus complee. Now as proved in Secion, he process X (..3). Then we have is exacly soluion of he equaion Corollary 3.3. The soluion of he fracional Ginzburg-Landau equaion dx =( X 3 +(α + σ )X ) d + σx db,

9 On a fracional sochasic Landau-Ginzburg equaion 35 is unique and given by X = σ σb+(α+ X e ) ( 1+X σ σbs+(α+ e )s) 1. Proof. The uniqueness of he soluion can be seen as follows: If X,1 and X, are wo limis of X ε in L, hen as ε. X, X ε X,1 + X ε X, X,1 References [1] E. Alòs, O. Maze and D. Nualar, Sochasic Calculus wih Respec o Fracional Brownian Moion wih Hurs Paramener less han 1.J. Sochasic Processes and heir Applicaions, 86 issue 1 (), [] L. Decreusefond and N. Savy, Filered Brownian moions as weak limi of filered Poisson processes. Bernoulli 11(), 5, [3] I. I. Gihman and A. V. Skorohod, Sochasic Differenial Equaions. Springer, 197. [4] I. S. Aranson and L. Kramer, The World of he Complex Ginzburg-Landau Equaion, Reviews of Modern Physics, Vol. 74 (), [5] S. C. Lim and V. M. Sihi, Asympoic properies of he fracional Brownian moion of Riemann-Liouville ype. Physics Leers A 6 (1995), [6] T. H. Thao, An Approximae Approach o Fracional Analysis for Finance, Nonlinear Analysis 7 (6), (available also online on ScienceDirec) Received: June, 9