Irreducibility of stochastic real Ginzburg Landau equation driven by α-stable noises and applications

Transcription

1 Bernoulli 232, 217, DOI: 1.315/15-BEJ773 Irreducibility of stochastic real Ginzburg Landau equation driven by α-stable noises and applications RAN WANG 1, JIE XIONG 2,* and LIU XU 2,**, 1 School of Mathematical Sciences, University of Science and Technology of China, efei, China. wangran@ustc.edu.cn 2 Department of Mathematics, Faculty of Science and Technology, University of Macau, Taipa, Macau. * jiexiong@umac.mo; ** lihuxu@umac.mo We establish the irreducibility of stochastic real Ginzburg Landau equation with α-stable noises by a maximal inequality and solving a control problem. As applications, we prove that the system converges to its equilibrium measure with exponential rate under a topology stronger than total variation and obeys the moderate deviation principle by constructing some Lyapunov test functions. Keywords: α-stable noises; exponential ergodicity; irreducibility; moderate deviation principle; stochastic real Ginzburg Landau equation 1. Introduction Consider the stochastic real Ginzburg Landau equation driven by α-stable noises on torus T := R/Z as follows: dx ξ 2 X dt X X 3 dt = dl t, 1.1 where X :, + T R and L t is an α-stable noise. It is known 22] that equation 1.1 admits a unique mild solution X in the càdlàg space almost surely. As α 3/2, 2, X is a Markov process with a unique invariant measure π see Section 2 below for details. By the uniqueness see 4], π is ergodic in the sense that T 1 lim X t dt = dπ, P-a.s. T T for all initial state x and all continuous and bounded functions. The irreducibility is a fundamental concept in stochastic dynamic system, and plays a crucial role in the research of ergodic theory. See, for instance, the classical work 1] and the books for stochastic infinite dimensional systems 4,15]. It is well known that one usually solves a control problem to prove the irreducibility for stochastic partial differential equations SPDEs driven by Wiener noises, see 3,4]. For SPDEs driven by α-stable noises, when the system is linear or Lipschitz, Priola and Zabczyk proved the Corresponding author ISI/BS

2 118 R. Wang, J. Xiong and L. Xu irreducibility in the same line 19]. owever, due to the discontinuity of trajectories and the lack of second moment, the control problem in 19] is much harder than those in the Wiener noises case. To our knowledge, there seem no other literatures about the irreducibility of stochastic systems with α-stable noises. In this paper, we prove that the system 1.1 is irreducible by following the spirit in 3] and 19]. Due to the non-lipschitz nonlinearity, the control problem in our setting is much harder and a maximal inequality is needed. The ergodicity of the system 1.1 has been proved in 22] in the sense that X converges to a unique invariant measure under the weak topology, but the convergence speed is not addressed. In this paper, thanks to the irreducibility and the strong Feller property established in 22], we prove that the system 1.1 converges to the invariant measure exponentially fast under a topology stronger than total variation by constructing a Lyapunov test function. Another application of our irreducibility result is to establish moderate deviation principle MDPof1.1. Thanks to 21], the MDP is obtained by verifying the same Lyapunov condition as above. Finally, we recall some literatures on the study of invariant measures and the long time behavior of stochastic systems driven by α-stable type noises. 17,18] studied the exponential mixing for a family of semi-linear SPDEs with Lipschitz nonlinearity, while 9] obtained the existence of invariant measures for 2D stochastic Navier Stokes equations forced by α-stable noises with α 1, 2. 23] proved the exponential mixing for a family of 2D SDEs forced by degenerate α-stable noises. For the long term behaviour about stochastic system drive by Lévy noises, we refer to 6,7,9,12 14] and the literatures therein. The paper is organized as follows. In Section 2, we first give a brief review of some known results about the existence and uniqueness of solutions and invariant probability measures for stochastic Ginzburg Landau equations. We will also present the main theorems in this section. In Section 3, we prove that the system X is irreducible. In the last section, we first recall some results about moderate deviations and exponential convergence for general strong Feller Markov processes, and then we prove moderate deviations and exponential convergence for X by constructing appropriate Lyapunov test functions. 2. Stochastic real Ginzburg Landau equations Let T = R/Z be equipped with the usual Riemannian metric, and let dξ denote the Lebesgue measure on T. For any p 1, let { L p 1/4 } T; R := x : T R; x L p := xξ 4 dξ <. Then { } := x L 2 T; R; xξdξ = T is a separable real ilbert space with inner product x,y := xξyξdξ x,y. T T

3 Irreducibility of stochastic real Ginzburg Landau equation 1181 For any x,let x := x L 2 = x,x 1/2. Let Z := Z \{}. It is well known that { ek ; e k = e i2πkξ },k Z is an orthonormal basis of. For each x, it can be represented by Fourier series x = k Z x k e k with x k C,x k = x k. Let be the Laplace operator on. It is well known that D = 2,2 T. In our setting, can be determined by the following relations: for all k Z, e k = γ k e k with γ k = 4π 2 k 2, with { 2,2 T = x ; x = x k e k, } γ k 2 x k 2 <. k Z k Z Denote Define the operator A σ with σ by A =, DA = 2,2 T. A σ x = k Z γ σ k x ke k, x D A σ, where {x k } k Z are the Fourier coefficients of x, and D A σ { := x : x = x k e k, } γ k 2σ x k 2 <. k Z k Z Given x DA σ, its norm is A σ x 1/2 := γ k 2σ x k 2. k Z Moreover, let V := D A 1/2 and x V := A 1/2 x. 2.1 Notice that V is densely and compactly embedded in.

4 1182 R. Wang, J. Xiong and L. Xu We shall study 1D stochastic Ginzburg Landau equation on T as the following { dxt + AX t dt = NX t dt + dl t, X = x, 2.2 where i the nonlinear term N is defined by Nu= u u 3, u. ii L t = k Z β k l k te k is an α-stable process on with {l k t} k Z being i.i.d. 1-dimensional symmetric α-stable process sequence with α>1, see 2]. Moreover, we assume that there exist some C 1,C 2 > so that C 1 γ β k β k C 2 γ β k with β> α 1. Let C>be a constant and let C p > be a constant depending on the parameter p. We shall often use the following inequalities 22]: A σ e At C σ t σ σ> t >; 2.3 Nx V C x V + x 3 V x V ; 2.4 ANx C 1 + x 2 V 1 + Ax 2 ; 2.5 x 4 L 4 x 2 V x 2 x V. 2.6 ere we consider a general E-valued càdlàg Markov process, { }, F t t, F, { Xt x }t,x E,P x x E whose transition probability is denoted by {P t x, dy} t, where := D, + ; E is the space of the càdlàg functions from, + to E equipped with the Skorokhod topology, Ft = σ {X s, s t} is the natural filtration. For all f bbe the space of all bounded measurable functions, define P t fx= P t x, dyfy for all t,x E. E For any t>, P t is said to be strong Feller if P t ϕ C b E for any ϕ bbe; P t is irreducible in E if P t 1 O x > for any x E and any non-empty open subset O of E. Definition 2.1. We say that a predictable -valued stochastic process X = Xt x is a mild solution to equation 2.2 if, for any t,x, it holds P-a.s.: X x t ω = e At x + e At s N X x s ω ds + e At s dl s ω. 2.7 The following existence and uniqueness results for the solutions and the invariant measure can be found in 22].

5 Irreducibility of stochastic real Ginzburg Landau equation 1183 Theorem ]. The following statements hold: 1 If α 1, 2 and β> α 1, for every x and ω a.s., equation 2.2 admits a unique mild solution Xx ω D, ; D, ; V. 2 X = Xt x t,x is a Markov process. If α 3/2, 2 and α <β< α are further assumed, the transition probability P t of X is strong Feller in for any t>. 3 If α 3/2, 2 and α 1 <β< 3 2 α 1, X admits a unique invariant measure, and the invariant measure is ported on V. Our first main result is the following theorem about the irreducibility. Theorem 2.3. Assume that α 1, 2 and β> α 1. For any initial value x, the Markov process X ={Xt x} t,x to the equation 2.2 is irreducible in. Remark 2.4. By the well-known Doob s theorem see 4], the strong Feller property and the irreducibility imply that X admits at most one unique invariant probability measure. This gives another proof to the uniqueness of invariant measure. As an application of our irreducibility result together with strong Feller property, we have the following exponential ergodicity under a topology stronger than total variation. Recall that in 22] by ergodicity we mean that X has a unique invariant measure under the weak topology. Theorem 2.5 below gives not only an ergodic theorem in stronger sense but also exponential convergence speed. Theorem 2.5. Assume that α 3/2, 2 and α 1 <β< 3 2 α 1. Let π be the unique invariant probability measure of X. Then there exist some positive constants M>1,ρ, 1, θ > satisfying that dπ <+, where x := M + x 2 1/2, and π is exponentially ergodic in the sense that P tfx f dπ θ x ρt x,t. 2.8 f Remark 2.6. Let B, be the Banach space of all real measurable functions f on such that fx f := x x < +. The exponential convergence 2.8 means that P t πf θ f ρ t, that is, P t has a spectral gap near its largest eigenvalue 1 in B. Let M b be the space of signed σ -additive measures of bounded variation on equipped with the Borel σ -field B.OnM b, we consider the topology σm b, bb,theso

6 1184 R. Wang, J. Xiong and L. Xu called τ -topology of convergence against all bounded Borel functions which is stronger than the usual weak convergence topology σm b, C b, see5], Section 6.2. Let L t A := 1 t δ Xs A ds for any measurable set A, where δ a is the Dirac measure at a. According to Corollary 2.5 in 21], the system X has the following exponential ergodicity. Corollary 2.7. Under the conditions of Theorem 2.5, the following results hold: a L t converges to π with an exponential rate w.r.t. the τ -topology. More precisely for any neighborhood N π of π in M b, τ, 1 lim K t + t log P x Lt / N π <. x K ere K means that K is a compact set in. b The process X is exponentially recurrent in the sense below: for any compact K in with πk >, there exists some λ > such that for any compact K in, E x exp λ τ K T < +, x K where τ K T = inf{t T ; X t K} for any T>. Another application of our irreducibility result together with strong Feller property is to establish MDP for the system 1.1. To this end, let us first briefly recall MDP as follows. Let bt : R +, + be an increasing function verifying lim bt =+, lim t bt =, 2.9 t t define M t := 1 bt δ Xs πds. 2.1 t Then moderate deviations of L t from its asymptotic limit π is to estimate P μ M t A, 2.11 where A is some measurable set in M b, τ, a given domain of deviation. ere P μ is the probability measure of the system X with initial measure μ. When bt = 1, this becomes an estimation of the central limit theorem; and when bt = t, it is exactly the large deviations. bt satisfying 2.9 is between those two scalings, called scaling of moderate deviations,see5]. Now we are at the position to state our MDP result.

7 Irreducibility of stochastic real Ginzburg Landau equation 1185 Theorem 2.8. In the context of Theorem 2.5, for any initial measure μ verifying μ < +, the measure P μ M t satisfies the large deviation principle w.r.t. the τ -topology with speed b 2 t and the rate function { Iν:= f dν 1 } 2 σ 2 f ; f bb ν M b, 2.12 where σ 2 1 f = lim t t Eπ fxs πf 2 ds 2.13 exists in R for every f B bb. More precisely, the following three properties hold: a1 for any a, {ν M b ; Iν a} is compact in M b, τ; a2 the upper bound for any closed set F in M b, τ, lim T 1 b 2 T log P μm T F inf F I; a3 the lower bound for any open set G in M b, τ, lim inf T 3. Irreducibility in 1 b 2 T log P μm T G inf G I. In this section, we shall prove that X ={X x t } t,x in the system 2.2 is irreducible in. Together with the strong Feller property established in 22], Theorem 6.1, this gives another proof to the existence of at most one invariant measure by classical Doob s theorem Irreducibility of stochastic convolution Let us first consider the following Ornstein Uhlenbeck process: dz t + AZ t dt = dl t, Z =, 3.1 where L t = k Z β k l k te k is an α-stable process on. It is well known that Z t = e At s dl s = z k te k, k Z where z k t = e γkt s β k dl k s. The following maximal inequality can be found in 22], Lemma 3.1.

8 1186 R. Wang, J. Xiong and L. Xu Lemma 3.1. For any T>, θ<β 2α 1 and all <p<α, we have E A θ Z t p CT p/α, t T where C depends on α, θ, β, p. The following lemma is concerned with the port of the distribution of {Z t } t T,Z T. Lemma 3.2. For any T>, <p<, the random variable {Z t } t T,Z T has a full port in L p,t]; V V. More precisely, for any φ L p,t]; V,a V,ε>, T P Z t φ t p V dt + Z T a V <ε >. Proof. First, by Lemma 3.1, wehavez L,T]; V, a.s. For any N N, let N be the ilbert space spanned by {e k } 1 k N, and let π N : N be the orthogonal projection. Notice that π N is also an orthogonal projection in V. Define π N = I π N, N = π N. By the independence of π N Z and π N Z, for any φ t L p,t]; V,a V,wehave T P T Z t φ t p V dt + Z T a V <ε T P πn Z t φ t p V dt + πn Z T a V < ε 2 p+1, π N Z t φ t p V dt + π N Z T a V < ε 2 p+1 T = P πn Z t φ t p V dt + πn Z T a V < T P π N Z t φ t p V dt + π N Z T a V < By the same argument as in the Section 4.2 of 19], we obtain ε 2 p+1 ε 2 p+1 T P π N Z t φ t p V dt + π N Z T a V < ε 2 p+1 >. To finish the proof, it suffices to show T P π N Z t φ t p V dt + π N Z T a V < ε 2 p+1 >..

9 Irreducibility of stochastic real Ginzburg Landau equation 1187 For any θ 1 2,β 2α 1, by Lemma 3.1 with p = 1 therein, the spectral gap inequality and Chebyshev inequality, we have for any η P π N Z V t η = 1 P π N Z V t >η t T t T 1 P π N A θ Z t >ηγ θ 1/2 N 1 P t T t T 1 C α,β,t η 1 γ 1/2 θ N, A θ Z t >ηγ θ 1/2 N where C α,β,t depends on α, β, T. By the previous inequality, as long as N depending on ε, p, φ is sufficiently large, we have P and T t T π N Z t V ε 2 2p+2 > π N φ t p V dt + π N a V < ε 2 2p+2. ence, T P π N Z t φ t p V dt + π N Z T a V < T P π N Z t p V dt + π N Z V T < ε 2 2p+2, T = P π N Z t p V dt + π N Z V T < ε 2 2p+2 >. ε 2 p+1 T π N φ t p V dt + π N a V < ε 2 2p+2 The proof is complete A control problem for the deterministic system Consider the deterministic system in, t xt + Axt = N xt + ut, x = x, 3.2 where u L 2,T]; V. By using the similar argument in the proof of Lemma 4.2 in 22], for every x = x,u L 2,T]; V, the system 3.2 admits a unique solution x

10 1188 R. Wang, J. Xiong and L. Xu C,T]; C,T]; V. Moreover, {xt} t,t ] has the following form: xt = e At x + e At s N xs ds + e At s us ds t,t]. 3.3 Next, we shall prove that the deterministic system is approximately controllable in time T>. Lemma 3.3. For any T>,ε >,a V, there exists some u L,T]; V such that the system 3.2 satisfies that xt a V <ε. Proof. We shall prove the lemma by the following three steps. Step 1. Regularization. For any t,t],letut = for all t,t ]. Then the system 3.2 admits a unique solution x C,t ]; C,t ]; Vwith the following form: xt = e At x + e At s N xs ds <t t. Step 2. Approximation at time T and linear interpolation. For any a V,ε>, there exists a constant θ> such that e θa a a V ε. Setting xt = t t T t e θa a + T t T t xt for all t t,t]. Then x C,T]; V.By3.2, we have ut = e θa a xt T t Axt N xt t t,t]. Step 3. It remains to show that u L,T]; V.By2.3, 2.4 and the constructions of {xt} t,t ] and {ut} t,t ] above, it is sufficient to show that Axt V. For any t t /2,t ], xt = e t t/3a xt /3 + e t sa N xs ds t /3 = e t t/2a xt /2 + e t sa N xs ds. t /2 By 2.3, we have for t> t 2, Axt = Ae t t /3A xt /3 + A Ae t t /3A xt /3 + t /3 t /3 e t sa N xs ds A 1/2 e t sa A 1/2 N xs ds 3.4 C 1 t t /3 1 xt /3 + C 1/2 t s 1/2 N xs V ds. t /3

11 Irreducibility of stochastic real Ginzburg Landau equation 1189 Since x C,t ]; V, t t /3,t ] xs V <. Together with 2.4 and 3.4, we obtain that Axt <. t t /2,t ] By the previous inequality and 2.5, we have A 3/2 xt = A3/2 e t/2a xt /2 + A 3/2 e t sa N xs ds t /2 A 3/2 e t/2a xt /2 + t /2 A 1/2 e t sa AN xs ds C 3/2 t /2 3/2 xt /2 + C 1/2 t s 1/2 AN xs ds t /2 C 3/2 t /2 3/2 xt /2 + C 1 + xs 2 V 1 + Axs 2 <, s t /2,t ] which means that Axt V. The proof is complete Irreducibility in Now we prove Theorem 2.3 by following the idea in 19], Theorem 5.4. Proof of Theorem 2.3. For any x,t >, we have X x t V a.s. by Theorem 2.2. Since X is Markov in, for any a,t >,ε>, To prove that P X x T a <ε = = it is sufficient to prove that for any T>, V V P X x T a <ε X x t = v P X x t dv P X v T t a <ε P X x t dv. P X x T a <ε >, P X x T a <ε > for all x V. Next, we prove the theorem under the assumption of the initial value x V in the following two steps.

12 119 R. Wang, J. Xiong and L. Xu Step 1. For any a,ε >, there exists some θ> such that e θa a V and a e θa a ε For any T >, by Lemma 3.3 and the spectral gap inequality, there exists some u L,T]; Vsuch that the system ẋ + Ax = Nx+ u, x = x, satisfies that xt e θa a xt e θa a V < ε Putting 3.5 and 3.6 together, we have xt a < ε Step 2. We shall consider the systems 3.8 and 3.9 as follows: and { ż + Az = u, z =, ẏ + Ay = Ny + z, y = x V, { dzt + AZ t dt = dl t, Z = ; dy t + AY t dt = NY t + Z t dt, Y = x V By the arguments in the proof of Lemma 4.2 in 22], for any x V,u L 2,T]; V,the systems 3.8 and 3.9 admit the unique solutions y, z C,T]; V 2 and Y,Z C,T]; V 2, a.s. Furthermore, denote For any t T, xt = yt + zt, X t = Y t + Z t t. Y t yt Y s ys 2 V ds = 2 = 2 2 Ys ys,nx s N xs ds Ys ys 2 ds + 2 Ys ys,z s zs ds Ys ys,x 3 s x3 s ds.

13 Irreducibility of stochastic real Ginzburg Landau equation 1191 Let us estimate the third term of the right-hand side. Denoting Y s = Y s ys and Z s = Z s zs, wehave Ys ys,x 3 s x3 s ds = = Ys, Y s + Z s + xs ] 3 x 3 s ds Ys, Y s + Z s ] Y s + Z s ] 2 xs + 3 Y s + Z s ]x 2 s ds = Ys, Y s Y s 2 Z s + 3 Y s Z s 2 + Z s 3 ds Ys, Y s Y s Z s + Z s 2] xs ds Ys, Y s + Z s ]x 2 s ds. Since 3 4 Y s Y s 3 xs + 3 Y s 2 x 2 s, from the above relation we have Ys ys,x 3 s x3 s ds Ys, 3 Y s 2 Z s + 3 Y s Z s 2 + Z s 3 ds Ys, 2 Y s Z s + Z s 2] xs ds Ys, Z s x 2 s ds + 1 Y s 4 4 L 4 ds. Using the following Young inequalities: for all y,z L 4 T; R, y,z = yξzξdξ T y4 ξ dξ + C z 4/3 ξ dξ, T 8 T y 2,z = y 2 ξzξ dξ T y4 ξ dξ + C z 2 ξ dξ, T 8 T y 3,z = y 3 ξzξ dξ T y4 ξ dξ + C z 4 ξ dξ, 8 T T

14 1192 R. Wang, J. Xiong and L. Xu and the ölder inequality, we further get Ys ys,x 3 s x3 s ds 1 8 6C 3C 1 8 6C 3C Y s 4 L 4 ds 7C Zs xs 2 L 2 ds 3C Zs x 2 s 4/3 L 4/3 ds Y s 4 L 4 ds 7C Z s 4 L 4 ds Z s 4 L 4 ds Z s 2 L 4 xs 2 L 4 ds 3C Z s 4/3 xs 8/3 L 4 L 4 ds. Zs 2 xs 4/3 L 4/3 ds Z s 8/3 xs 4/3 L 4 L 4 ds Since xt = yt + zt C,T]; V,by2.6, there exists a constant C T such that s,t ] ys + zs L 4 s,t ] Consequently, there is some constant C T > satisfying that Y t yt Y s ys 2 V ds 3 Y s ys 2 ds + Z s zs 2 ds ys + zs 1/2 ys + zs 1/2 V C T. + C Zs T zs 4 L 4 + Zs zs 2 L 4 + Zs zs 8/3 L 4 + Zs zs 4/3 L 4 ds ds. Therefore, by the spectral gap inequality and Gronwall s inequality, we have Y T yt 2 C T i T Z s zs i ds, 3.1 V where := {4/3, 2, 8/3, 4}. This inequality, together with Lemma 3.2, 3.7, implies P X T a <ε = P Y T yt + Z T zt + xt a <ε

15 Irreducibility of stochastic real Ginzburg Landau equation 1193 P YT yt ε/4, ZT zt ε/4, xt a <ε/2 = P Y T yt ε/4, Z T zt ε/4 T P Z s zs i V ds + Z T zt V C T,ε >. i The proof is complete. 4. The proofs of Theorems 2.5 and Several general results for strong Feller Markov processes In this subsection, we recall some general results about moderate deviations and exponential convergence for general strong Feller Markov processes, borrowed from 21]. We say that a measurable function f : R belongs to the extended domain D e L of the generator L of P t, if there is a measurable function g : R so that g X s ds < +, t >, P x -a.s. and fx t fx gx s ds, t is a càdlàg P x -local martingale for all x. In that case, g := Lf. Theorem ], Theorem 5.2c, or 21], Theorem 2.4. Assume that the process X t is strong Feller, irreducible and aperiodic see 11] for definition, that is the case if P T,K> over for some compact K verifying πk >. If there are some continuous function 1 D e L, compact subset K and constants ε, C > such that then there is a unique invariant probability measure π satisfying dπ <+, L ε1 K c C1 K, 4.1 and there are some constants θ>and <ρ<1 such that for all t, P tfz f dπ θ z ρt, z. 4.2 f For the measure-valued process M t defined in 2.1, we have the following large deviations result.

16 1194 R. Wang, J. Xiong and L. Xu Theorem ], Theorem 2.6. Assume that the process X t is strong Feller, irreducible, aperiodic and satisfies 4.1. For any initial measure μ verifying μ < +, the measure P μ M t satisfies the large deviation principle w.r.t. the τ -topology with the speed b 2 t and the rate function { Iν:= f dν 1 } 2 σ 2 f ; f bb ν M b, 4.3 where σ 2 1 f = lim t t E π exists in R for every f B bb The proofs of main results fxs πf 2 ds 4.4 In this subsection, we shall prove Theorems 2.5 and 2.8 based on the above theorems. The main technique is to construct a suitable Lyapunov test function. Proofs of Theorems 2.5 and 2.8. By Theorems 2.2 and 2.3, the system 2.2 is strong Feller, irreducible and aperiodic in. Indeed, as the invariant measure π is ported on V, there exists a bounded closed ball F V satisfying πf >. Notice that F is compact in. Since the system X is strong Feller and irreducible in,by4], Theorem 4.2.1, the invariant measure π is equivalent to all measures P t x,, for all x,t >. Consequently, P t x, F > for all x,t >, which implies that the system is aperiodic. By Theorems 4.1 and 4.2, we now construct a suitable Lyapunov function satisfying 4.1. Take x := M + x 2 1/2, 4.5 where M is a large constant to be determined later. By Lemma 4.3 below, we have D e L. Recall x m = π m x,wehave L x m = Ax m,d x m + N x m,d x m + x m + β i y i e i x m β i y i D ei x m] νdy i i m + i m y i 1 y i >1 =: J m 1 + J m 2 + J m 3 + J m 4. x m + β i y i e i x m ] νdy i 4.6 ere D ei x m := x i x m, D x m := Dei x m e i = xm x m. i m

17 Irreducibility of stochastic real Ginzburg Landau equation 1195 For the first term, using the integration by parts formula, we have Ax m,d x m = Ax m x m, x m = xm 2 V x m. 4.7 For the second term, by 2.5 in 22] note that Nx here equals Nx in 22], we have For any h, N x m,d x m = N x m, x m x m 1 4 x m. 4.8 h, D 2 x m h h m 2 = M + x m hm,x m 2 2 M + x m 2 h 2 3/2 M + x m 2 This inequality, together with Taylor s formula, implies that x m + β i y i e i x m β i y i D ei x m β 2 i y2 i M + x m 2.. Thus, for the third term, we have i m y i 1 = i m x m + β i y i e i x m β i y i D ei x m νdy i y i 1 β 2 i y2 i M + x m 2 νdy i i m β2 i y 1 α dy M + x m 2 y 1 C α i m = β2 i C α 2 α M + x m 2 2 i Z < βi 2 C α 2 α M + x m 2 < +, where ν is the Lévy measure of 1-dimensional α-stable process and we have used the assumption of β i in ii in the last inequality. By Taylor s formula again, there exists x m satisfying that x m + β i y i e i x m = D ei x m,β i y i e i = x i β i y i M + x m 2 β i y i.

18 1196 R. Wang, J. Xiong and L. Xu For the fourth term, we have i m y i >1 i m = i m x m + β i y i e i x m ] νdy i y i >1 y i >1 β i y i νdy i 4.1 β i y i C α y i 1+α dy i = 2 i m β i < 2 i Z β i C α α 1 C α α 1 < +, where we have used the assumption of β i in ii again. Putting together, we obtain that for any x, Let L xm m x m = J 1 + J 2 m + J 3 m + J 4 m x m xm 2 V 1 M + x m 2 4M + x m 2 2 i Z βi 2 C α 2 αm + x m i Z β i. C α α 1 M + x m 2 K := { x V ; x 2 V M}. Then K is compact in. By 3 in Theorem 2.2, choose M large enough such that πk > and 1 4M + 2 i Z βi 2 C α 2 αm + 2 i Z β i C α α 1 M 1 4 x Since lim m x m = x and lim m L x m has limit for x V, by the closable property of L, we immediately get L x = lim m L x m and thus For any x K c,by4.11 and 4.12, we have L x x 1, x K L xm x m xm 2 V M + x m

19 Irreducibility of stochastic real Ginzburg Landau equation 1197 This implies and L x x xm 2 V M + x m x V K c 4.14 L x x = x \ V K c Putting together, we immediately obtain L x x K c K. The proof is complete. Lemma 4.3. For defined in 4.5, we have D e L. Before proving D e L, let us first briefly review some well known facts about α-stable process for using Itô formula. Let {l j t} j 1 be a sequence of i.i.d. 1-dimensional α-stable processes. The Poisson random measure associated with l j t is defined by N j t, Ɣ := s,t] 1 Ɣ lj s l j s t>, Ɣ B R \{}. By Lévy Itô s decomposition cf. 1], page 126, Theorem , one has l j t = xñ j t, dx + xn j t, dx, x 1 x >1 where Ñ j is the compensated Poisson random measure defined by Ñ j t, Ɣ = N j t, Ɣ tνɣ. Proof of Lemma 4.3. The proof follows from 8], Section 3, in spirit. Let T>beanarbitrary but finite number, we shall consider the stochastic system in,t]. Consider the Galerkin approximation of 2.2: dx m t + AX m t dt = N m X m t dt + dl m t, X m = xm, 4.16 where Xt m = π m X t, N m Xt m = π m NXt m ], L m t = k m β kl k te k, π m is the orthogonal projection defined in the proof of Lemma 3.2. By a standard argument 2,16], for all x W with W = or W = V we have E t T X m t x m W ] C W x, T, 4.17

20 1198 R. Wang, J. Xiong and L. Xu lim E X m m t x m X t x ] W =, 4.18 t T where C W x, T > is finite. Write u := M + u 2 1/2 for all u, it follows from Itô formula 1] that X m t x m + I1 m t I 2 m t = I 3 m t + I 4 m t, 4.19 where I1 m t := Xs m 2 V M + Xs m 2 t := I m 2 I m 3 I m 4 j m t := j m t := j m R y 1 y >1 ds 1/2 Xs m,nxm s M + Xs m ds, 2 1/2 X m s + yβ j e j X m s X m s,yβ j e j M + X m s 2 1/2 1 { y 1} X m s + yβ j e j X m s ]Ñ j ds,dy, X m s + yβ j e j X m s ]Ñ j ds,dy. ] νdyds, By a Taylor expansion argument similar to 4.9 and 4.1 below, for all T>wehave E I m 2 t ] CT, E E t T t T t T t T I3 m t 2] CT, I4 m t ] CT. Moreover, by a Taylor expansion argument similar to 4.9 and 4.1 below again, we get that as m 1,m 2, E I m 1 2 t I m 2 2 t ], E E t T t T I m 1 3 t I m 2 3 t 2], I m 1 4 t I m 2 4 t p], for all 1 p<α. ence, there exist I 2, I 3 and I 4 such that lim E I m m 2 t I 2t ] =, 4.2 t T

21 Irreducibility of stochastic real Ginzburg Landau equation 1199 lim I m m 3 t I 3 t 2] =, 4.21 t T lim E I m m 4 t I 4t p] =, 4.22 t T where I 2, I 3 and I 4 have the same forms as I m 2, I m 3 and I m 4 but with i m replaced by i Z and X m replaced by X. It is also easy to verify that I 3 is an L 2 martingale and that I 4 is an L 1 martingale. Next we shall show below, taking a subsequence if necessary, that lim m I m 1 t = I 1t t T,a.s., 4.23 where I 1 t has the same form as I m 1 t but with Xm replaced by X. Collecting , taking a subsequence if necessary and letting m in 4.19, we obtain X t x+ I 1 t I 2 t = I 3 t + I 4 t Since I 3 and I 4 are L 2 and L 1 martingales respectively, taking gt = X s 2 V M + X s 2 + j Z R ds + 1/2 X s,nx s M + X s 2 ds 1/2 X s + yβ j e j X s X s,yβ j e j M + X s 2 1/2 1 { y 1} ] νdyds, we immediately verify that D e L for t,t]. Since T>is arbitrary, D e L for t,. It remains to prove Taking a subsequence if necessary and letting m in 4.19, by Fatou lemma and the fact x,nx 1 4 from 22]wehave E t,t ] This implies It is easy to check M + Xt 2 1/2 ] X m s 2 V M+ X m s 2 T + E X s 2 ] V M + X s 2 ds M + x 2 1/2 + CT + CT 1/2. 1/2 X s 2 V M + X s 2 ds < t,t], a.s /2 1/2 is increasing in m for every s> and Xs m 2 V M + Xs m X s 2 V 2 1/2 M + X s 2, s >. 1/2

22 12 R. Wang, J. Xiong and L. Xu ence, by 4.18 and the Lesbegue dominated convergence theorem, taking a subsequence if necessary, we get lim m X m s 2 V M + Xs m ds = 2 1/2 Furthermore, observe that x,nx xξ 2 dξ + xξ 4 dξ T T X s 2 V M + X s 2 ds a.s /2 x 2 + x 2 x 2 x 2 + C x 2 V x 2, where the last inequality is by Sobolev embedding. Note that for all s,t] Xs m,nxm s M + Xs m 2 1/2 t T Xm t C X s m 2 V M + Xs m 2 1/2 t T Xm t C X s 2 V M + X s 2 1/2 a.s. ence, taking a subsequence if necessary, by 4.17, 4.18 and 4.25 with the Lesbegue dominated convergence theorem, we obtain lim m Xs m,nxm s M + Xs m ds = 2 1/2 X s,nx s M + Xs m ds a.s /2 Combining 4.26 and 4.27, we immediately get the desired equation Acknowledgments The authors are grateful to the anonymous referees for constructive comments and corrections. R. Wang thanks the Faculty of Science and Technology, University of Macau, for finance port and hospitality. e was ported by Natural Science Foundation of China , and the Fundamental Research Funds for the Central Universities WK148. J. Xiong was ported by Macao Science and Technology Fund FDCT 76/212/A3 and Multi-Year Research Grants of the University of Macau Nos. MYRG FST and MYRG FST. L. Xu is ported by the grants: SRG FST, MYRG FST and Science and Technology Development Fund, Macao S.A.R FDCT 49/214/A1. All of the three authors are ported by the research project RDAO/RTO/386-SF/214. References 1] Applebaum, D. 29. Lévy Processes and Stochastic Calculus, 2nd ed. Cambridge Studies in Advanced Mathematics 116. Cambridge: Cambridge Univ. Press. MR25128

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