Freidlin-Wentzell Type Large Deviation Principle for Multi-Scale Locally Monotone SPDEs
aa r X i v : . [ m a t h . P R ] F e b Freidlin-Wentzell Type Large Deviation Principle forMulti-Scale Locally Monotone SPDEs
Wei Hong a , Shihu Li b , Wei Liu b,c ∗ a. Center for Applied Mathematics, Tianjin University, Tianjin 300072, China b. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China c. Research Institute of Mathematical Sciences, Jiangsu Normal University, Xuzhou 221116, China
Abstract.
This work is concerned with the Freidlin-Wentzell type large devia-tion principle for a family of multi-scale quasi-linear and semi-linear stochasticpartial differential equations (SPDEs) with small multiplicative noise under thegeneralized variational setting, which extend several existing works to the multi-scale process. Employing the weak convergence method developed by Dupuisand Ellis [7] and Khasminskii’s time discretization approach [18], the Laplaceprinciple for SPDEs will be derived, which is equivalent to the large deviationprinciple. In particular, in this paper, we do not assume any compactness ofthe embedding on the Gelfand triple we considered in order to deal with thecase of bounded and unbounded domains in some concrete models. Our mainresults are applicable to a wide family of SPDEs such as stochastic porous mediaequations, stochastic fast-diffusion equations, stochastic 2D hydrodynamical typemodels, stochastic p-Laplace equations, stochastic power law fluid equations andstochastic Ladyzhenskaya models.
Keywords:
SPDE; Multi-scale; Large deviation principle; Porous media equa-tion; Fast-diffusion equation; Navier-Stokes equation; Variational setting.
Mathematics Subject Classification (2010):
The large deviation principle (LDP) mainly investigates the asymptotic property of re-mote tails of a family of probability distributions, which is one of important topic in theprobability theory and has been widely applied to the fields that thermodynamics, statistics,information theory, engineering, etc. We refer the interested readers to the classical bookby Varadhan [37] for this theory with some of important applications. Owing to the workof Freidlin and Wentzell [11], the well-known small perturbation type (also called Freidlin-Wentzell type) large deviations for the stochastic differential equations (SDEs) has beenextensively studied in the recent decades, one might refer to [1, 33] and reference within.There are numerous results concerning the LDP for SDEs/SPDEs with small perturbationin the literature making use of different methods. For instance, Chow [5] considered the LDPfor a class of semilinear parabolic equations within a framework of the semigroup approach,R¨ockner et al. [30] established the LDP for stochastic generalized porous media equations in ∗ Corresponding author: [email protected]
Let ( U, h· , ·i U ) and ( H i , h· , ·i H i ), i = 1 ,
2, be the separable Hilbert spaces, and H ∗ i thedual space of H i . Let V i , i = 1 ,
2, denote a reflexive Banach space such that the embedding V i ⊂ H i is continuous and dense. Identifying H i with its dual space by the Riesz isomorphism,we have a so-called Gelfand triple V i ⊂ H i ( ∼ = H ∗ i ) ⊂ V ∗ i . The dualization between V i and V ∗ i is denoted by V ∗ i h· , ·i V i . Moreover, it is easy to see that V ∗ i h· , ·i V i | H i × V i = h· , ·i H i , i = 1 , . Let L ( U, H i ) be the space of all Hilbert-Schmidt operators from U to H i .Let T > A : V → V ∗ , F : H × H → H , G : V → L ( U, H ) , F : H × V → V ∗ , G : H × V → L ( U, H ) , we consider the following two-time-scale SEEs, dX ǫ,αt = (cid:2) A ( X ǫ,αt ) + F ( X ǫ,αt , Y ǫ,αt ) (cid:3) dt + √ ǫG ( X ǫ,αt ) dW t ,dY ǫ,αt = 1 α F ( X ǫ,αt , Y ǫ,αt ) dt + 1 √ α G ( X ǫ,αt , Y ǫ,αt ) dW t ,X ǫ,α = x, Y ǫ,α = y, (2.1)where { W t } t ∈ [0 ,T ] is an U -valued cylindrical Wiener process defined on a complete filteredprobability space (Ω , F , F t ≥ , P ) (i.e. the path of W take values in C ([0 , T ]; U ), where U is another Hilbert space in which the embedding U ⊂ U is Hilbert–Schmidt).We impose that the coefficients of system Eq. (2.1) satisfy the following two hypothesises,respectively: Hypothesis 2.1
For the slow component of Eq. (2.1), we assume that there are some con-stants γ > , β ≥ , θ > and K, C > such that for all u, v, w ∈ V , u , u ∈ H and v , v ∈ H ,( A1 ) (Hemicontinuity) The map λ V ∗ h A ( u + λv ) , w i V is continuous on R .( A2 ) (Local monotonicity and Lipschitz) V ∗ h A ( u ) − A ( v ) , u − v i V + k G ( u ) − G ( v ) k L ( U,H ) ≤ − θ k u − v k γ V + ( K + ρ ( v )) k u − v k H , where ρ : V → [0 , + ∞ ) is a measurable and locally bounded function on V and satisfies ρ ( v ) ≤ C (1 + k v k γ V )(1 + k v k βH ) . Moreover, k F ( u , v ) − F ( u , v ) k H ≤ C (cid:0) k u − u k H + k v − v k H (cid:1) . and k G ( u ) − G ( v ) k L ( U,H ) ≤ C k u − v k H . ( A3 ) (Growth) k A ( u ) k γ γ − V ∗ ≤ C (1 + k u k γ V )(1 + k u k βH ) , Remark 2.1
By ( A2 ) and ( A3 ), the coercivity of A and G is easily obtained as V ∗ h A ( u ) , u i V + k B ( u ) k L ( U,H ) ≤ − θ k u k γ V + K (1 + k u k H ) . Hypothesis 2.2
For the fast component of Eq. (2.1), we assume that there are some con-stants γ > , κ, θ ≥ and C, L G > such that for all u, v, w ∈ V , u , v ∈ H ,( H1 ) (Hemicontinuity) The map λ V ∗ h F ( u + λv ) , w i V is continuous on R . H2 ) (Monotonicity and Lipschitz) V ∗ h F ( u , u ) − F ( u , v ) , u − v i V + k G ( u , u ) − G ( u , v ) k L ( U,H ) ≤ − κ k u − v k H . Moreover, V ∗ h F ( u , u ) − F ( v , u ) , v i V ≤ C k u − v k H k v k H (2.2) and k G ( u , u ) − G ( v , v ) k L ( U,H ) ≤ C k u − v k H + L G k u − v k H . ( H3 ) (Coercivity) V ∗ h F ( u , v ) , v i V ≤ C k v k H − θ k v k γ V + C (1 + k u k H ) . ( H4 ) (Growth) k F ( u , v ) k γ γ − V ∗ ≤ C (1 + k v k γ V + k u k H ) and sup y ∈ V k G ( x, y ) k L ( U,H ) ≤ C (1 + k x k H ) . (2.3)The definition of variational solution to Eq. (2.1) is presented as follows. Definition 2.1
For any ǫ, α > , we call a continuous H × H -valued ( F t ) t ≥ -adaptedprocess ( X ǫ,αt , Y ǫ,αt ) t ∈ [0 ,T ] is a solution of the system (2.1), if for its dt × P -equivalent class ( ˆ X ǫ,αt , ˆ Y ǫ,αt ) t ∈ [0 ,T ] satisfying ˆ X ǫ,α ∈ L γ (cid:0) [0 , T ] × Ω , dt × P ; V (cid:1) ∩ L (cid:0) [0 , T ] × Ω , dt × P ; H (cid:1) , ˆ Y ǫ,α ∈ L γ (cid:0) [0 , T ] × Ω , dt × P ; V (cid:1) ∩ L (cid:0) [0 , T ] × Ω , dt × P ; H (cid:1) , where γ , γ is the same as defined in ( A2 ) and ( H3 ), respectively, and P -a.s. d ¯ X ǫ,αt = (cid:2) A ( ¯ X ǫ,αt ) + F ( ¯ X ǫ,αt , Y ǫ,αt ) (cid:3) dt + √ ǫG ( ¯ X ǫ,αt ) dW t ,d ¯ Y ǫ,αt = 1 α F ( ¯ X ǫ,αt , ¯ Y ǫ,αt ) dt + 1 √ α G ( ¯ X ǫ,αt , ¯ Y ǫ,αt ) dW t , here ( ¯ X ǫ,α , ¯ Y ǫ,α ) is an V × V -valued progressively measurable dt × P -version of ( ˆ X ǫ,α , ˆ Y ǫ,α ) . Lemma 2.1 ([27, Theorem 2.3]) Suppose that the assumptions ( A1 )-( A3 ) and ( H1 )-( H4 )hold. For each ǫ, α > and starting points ( x, y ) ∈ H × H , Eq. (2.1) has a unique solution ( X ǫ,αt , Y ǫ,αt ) t ∈ [0 ,T ] . Let us first recall some standard definitions and results of large deviation theory. Let { X ε } be a family of random variables defined on a probability space (Ω , F , P ), which takevalues in some Polish space E . Roughly speaking, the large deviation theory concerns theexponential decay of the probability measures of certain kinds of extreme or tail events. Therate of such exponential decay is expressed by the “rate function”. Definition 2.2 (Rate function) A function I : E → [0 , + ∞ ) is called a rate function if I islower semicontinuous. Moreover, a rate function I is called a good rate function if the levelset { x ∈ E : I ( x ) ≤ K } is compact for each constant K < ∞ . efinition 2.3 (Large deviation principle) The sequence { X ε } is said to satisfy the large de-viation principle on E with rate function I if the following lower and upper bound conditionshold,(i) (Lower bound) For any open set G ⊂ E : lim inf ε → ε log P ( X ε ∈ G ) ≥ − inf x ∈ G I ( x ) . (ii) (Upper bound) For any closed set F ⊂ E : lim sup ε → ε log P ( X ε ∈ F ) ≥ − inf x ∈ F I ( x ) . The weak convergence method is based on the equivalence between the LDP and theLaplace principle, defined as follows.
Definition 2.4 (Laplace principle) The sequence { X ε } is said to satisfy the Laplace princi-ple on E with a rate function I if for each bounded continuous real-valued function h definedon E , we have lim ε → ε log E (cid:26) exp (cid:20) − ε h ( X ε ) (cid:21)(cid:27) = − inf x ∈ E { h ( x ) + I ( x ) } . Lemma 2.2 (Varadhan’s Lemma [37]) Let E be a Polish space and an E -valued randomsequence { X ε } fulfill LDP with rate function I . Then { X ε } fulfills the Laplace principle on E with the same rate function I . Lemma 2.3 (Bryc’s converse [8]) The Laplace principle implies the LDP with the samerate function.
Combining Lemma 2.2 and 2.3 yields that if E is a Polish space and I is a good ratefunction, then the large deviation principle and Laplace principle are equivalent.Let A = (cid:26) φ : φ is U -valued F t -predictable process and Z T k φ s ( ω ) k U ds < ∞ P - a.s. (cid:27) , and S M = (cid:26) φ ∈ L ([0 , T ] , U ) : Z T k φ s k U ds ≤ M (cid:27) . It is well-known that the set S M is a compact metric space under the metric d ( u, v ) := P ∞ i =1 12 i (cid:12)(cid:12)(cid:12) R T h u ( t ) − v ( t ) , e i ( t ) i U ds (cid:12)(cid:12)(cid:12) , here { e i } i ≥ are an orthonormal basis of L ([0 , T ]; U ),then S M endowed with the weak topology is a Polish space (hereafter, we always refer to theweak topology on S M unless stated otherwise). We also define A M = { φ ∈ A : φ · ( ω ) ∈ S M , P - a.s. } . Let E be a Polish space, for any ε >
0, suppose G ε : C ([0 , T ]; U ) → E is a measurablemap and X ε = G ε ( W · ).We now formulate the sufficient condition for the Laplace principle (equivalently, theLDP) of X ε as ε → Condition (A) : There exists a measurable map G : C ([0 , T ]; U ) → E for which thefollowing conditions hold: 6i) Let { φ ε : ε > } ⊂ A M for some M < ∞ . If φ ε converge to φ in distribution as S M -valued random elements, then G ε (cid:18) W · + 1 √ ε Z · φ εs ds (cid:19) → G (cid:18)Z · φ s ds (cid:19) in distribution as ε → M < ∞ , the set K M = (cid:26) G (cid:18)Z · φ s ds (cid:19) : φ ∈ S M (cid:27) is a compact subset of E .Budhiraja and Dupuis [2] proved the following result for the Laplace principle (equiva-lently, the LDP). Lemma 2.4 [2, Theorem 4.4] If X ε = G ε ( W · ) and Condition (A) holds, then the family { X ε } satisfies the Laplace principle (hence LDP) on E with the good rate function II ( f ) = inf { φ ∈ L ([0 ,T ]; U ): f = G ( R · φ s ds ) } (cid:26) Z T k φ s k U ds (cid:27) , (2.4) where infimum over an empty set is taken as + ∞ . Remark 2.2
It should be mentioned that
Condition (A) (i) characterize the weak con-vergence of a certain family of random elements and is the most important point of so-calledweak convergence method to investigate the LDP. The
Condition (A) (ii) implies that thelevel sets of the rate function are compact.
As is well-known (cid:0) C ([0 , T ]; H ) ∩ L γ ([0 , T ]; V ) , d ( · , · ) (cid:1) is a Polish space with the metric d ( f, g ) := sup t ∈ [0 ,T ] k f t − g t k H + (cid:18)Z T k f t − g t k γ V dt (cid:19) γ . (2.5)According to the Yamada–Watanabe theorem, there exists a Borel-measurable function G ε : C ([0 , T ]; U ) → C ([0 , T ]; H ) ∩ L γ ([0 , T ]; V ) (2.6)such that X ǫ,α · = G ε ( W · ), P - a.s. , here X ǫ,α is the unique solution to slow equation of system(2.1).Consider the following skeleton equation: d ¯ X φt dt = (cid:2) A ( ¯ X φt ) + ¯ F ( ¯ X φt ) (cid:3) + G ( ¯ X φt ) φ t , ¯ X φ = x, (2.7)where φ ∈ L ([0 , T ]; U ) and ¯ F ( x ) := R H F ( x, y ) µ x ( dy ) , x ∈ H for µ x being the uniqueinvariant measure of the Markov semigroup to the frozen equation (see Eq. (3.1)). Theexistence and uniqueness of solution to Eq. (2.7) for any φ ∈ L ([0 , T ]; U ) will be provedin the next section (see Lemma 3.3). Furthermore, we define a map G : C ([0 , T ]; U ) → C ([0 , T ]; H ) ∩ L γ ([0 , T ]; V ) by G (cid:16) Z · φ s ds (cid:17) := ¯ X φ · . Now we are in a position to state the main result of this work.7 heorem 2.1
Assume that Hypothesis 2.1 and 2.2 hold. If κ > L G defined in ( H2 ) andfor each ǫ, α > , lim ǫ → α ( ǫ ) = 0 and lim ǫ → αǫ = 0 . (2.8) Then as ǫ → , { X ǫ,α : ǫ > } satisfies the LDP on C ([0 , T ]; H ) ∩ L γ ([0 , T ]; V ) with thegood rate function I given by (2 . for f ∈ C ([0 , T ]; H ) ∩ L γ ([0 , T ]; V ) . Remark 2.3
This theorem can not be applied to the stochastic fast-diffusion equation andstochastic p-Laplace equation for the case of < p < directly since the condition ( A2 ) doesnot satisfy. However, if we replace ( A2 ) by the classical local monotonicity and coercivityas in [24] as follows:( A4 ) V ∗ h A ( u ) − A ( v ) , u − v i V + k G ( u ) − G ( v ) k L ( U,H ) ≤ ( K + ρ ( v )) k u − v k H , V ∗ h A ( u ) , u i V + k G ( u ) k L ( U,H ) ≤ − θ k u k γ V + K (1 + k u k H ) . Then the LDP for multi-scale stochastic fast-diffusion equation and stochastic p-Laplace equa-tion in the case of < p < can be proved on C ([0 , T ]; H ) . Theorem 2.2
Assume that ( A1 ), ( A3 ), ( A4 ) and Hypothesis 2.2 hold. If κ > L G andthe condition (2.8) holds, then as ǫ → , { X ǫ,α : ǫ > } satisfies the LDP on C ([0 , T ]; H ) with the good rate function I given by (2 . for f ∈ C ([0 , T ]; H ) .Proof Since we only use ( A2 ) to verify the additional convergence in L α ([0 , T ]; V ), if weconcern the LDP on C ([0 , T ]; H ), follow the similar arguments as in Theorem 2.1, we arein a position to get this theorem directly. Since the proof is just a small modification toTheorem 2.1, we omit the details to save space. (cid:3) The LDP for { X ǫ,α : ǫ > } in C ([0 , T ]; H ) ∩ L γ ([0 , T ]; V ) can be proved by thefollowing steps. We will show the existence and uniqueness of solution to the skeletonequation and controlled stochastic equations in C ([0 , T ]; H ) ∩ L γ ([0 , T ]; V ), which help usto verify two important results on the compactness of the level sets and the weak convergenceof the stochastic controlled equations. The classical Khasminskii method based on timediscretization will be applied to the proof of the weak convergence.Throughout this paper, we use C p ,p , ··· to denote a generic positive constant whose valuemay change from line to line, but depends only on the designated variables p , p , · · · . In this section, we consider the frozen equations corresponding to the fast equation ofsystem (2.1) for any fixed slow component x ∈ H and the skeleton equation (2.7) associatedwith the slow equation of system (2.1). We will present the existence and uniqueness ofinvariant probability measure and the associated exponential ergodicity to the frozen equa-tions in order to define the coefficient ¯ F in the skeleton equation. Then we consider thestochastic control problem with respect to the system (2.1) and give some crucial lemmaswhich will be used frequently throughout this paper.8 .1 Frozen and skeleton equations For each fixed slow component x ∈ H , the frozen equation to the fast equation of system(2.1) is given by ( dY t = F ( x, Y t ) dt + G ( x, Y t ) d f W t ,Y = y ∈ H , (3.1)where f W t is a cylindrical Wiener process in Hilbert space U , which is independent of W t .It is obvious that following from [24, Theorem 4.2.4], by Hypothesis 2.2, there is a uniquesolution denoted by Y x,yt to Eq. (3.1), which is a homogeneous Markov process.Let { P xt } t ≥ denote the Markov transition semigroup of process { Y x,yt } t ≥ , i.e. for anybounded measurable map f on H , set P xt f ( y ) = E f ( Y x,yt ) , y ∈ H , t > . According to [24, Theorem 4.3.9], it leads to the following exponential ergodicity directly.
Lemma 3.1
There is a constant
C > such that for all Lipschitz function f : H → R , itfollows that (cid:12)(cid:12)(cid:12) P xt f ( y ) − Z H f ( z ) µ x ( dz ) (cid:12)(cid:12)(cid:12) ≤ C (1 + k x k H + k y k H ) e − ̺t k f k Lip , (3.2) here µ x is a unique invariant probability measure of { P xt } t ≥ , and k f k Lip is the Lipschitzconstant of map f . Lemma 3.2
There is a constant
C > such that for any x , x ∈ H and y ∈ H we have sup t ≥ E k Y x ,yt − Y x ,yt k H ≤ C k x − x k H . Proof
Taking Z x ,x t := Y x ,yt − Y x ,yt , which satisfies the following SPDE dZ x ,x t = (cid:2) F ( x , Y x ,yt ) − F ( x , Y x ,yt ) (cid:3) dt + (cid:2) G ( x , Y x ,yt ) − G ( x , Y x ,yt ) (cid:3) d f W t , Z = 0 . (3.3)Applying Itˆo’s formula to k · k H (see [24, Theorem 4.2.5]) that k Z x ,x t k H = 2 Z t V ∗ h F ( x , Y x ,ys ) − F ( x , Y x ,ys ) , Z x ,x s i V ds + Z t k G ( x , Y x ,ys ) − G ( x , Y x ,ys ) k L ( U,H ) ds +2 Z t h (cid:2) G ( x , Y x ,ys ) − G ( x , Y x ,ys ) (cid:3) d f W s , Z x ,x s i H . Taking expectation and differentiating with respect to t , then the condition ( H2 ) leads to d E k Z x ,x t k H dt = 2 E h V ∗ h F ( x , Y x ,yt ) − F ( x , Y x ,yt ) , Z x ,x t i V i + 2 E h V ∗ h F ( x , Y x ,yt ) − F ( x , Y x ,yt ) , Z x ,x t i V i + E h k G ( x , Y x ,yt ) − G ( x , Y x ,yt ) k L ( U,H ) i ≤ − κ E k Z x ,x t k H + ε E k Z x ,x t k H + 2 L G E k Z x ,x t k H + C ε k x − x k H = − ( κ − L G − ε ) E k Z x ,x t k H + C ε k x − x k H , ε small enough and usingthe comparison theorem yields that for all t > E k Z x ,x t k H ≤ C k x − x k H Z t e − η ( t − s ) ds ≤ C η k x − x k H , where we denote η := κ − L G − ε >
0, which implies the assertion. (cid:3)
Now we recall ¯ X φt defined in the skeleton equation (2.7). The existence and uniquenessof solution to Eq. (2.7) are considered in the following lemma, moreover, some importantenergy estimates for the skeleton equation are also derived in order to prove the main resultsstated in this work. Lemma 3.3
Suppose that Hypothesis 2.2 holds. For each x ∈ H and φ ∈ L ([0 , T ]; U ) ,there exists a unique solution to Eq. (2.7) fulfilling sup φ ∈ S M n sup t ∈ [0 ,T ] k ¯ X φt k H + θ Z T k ¯ X φt k γ V dt o ≤ C (1 + k x k H ) , (3.4) here C > is a constant which is independent of φ .Proof We shall split the proof into the following two steps.
Step 1 : In order to prove the well-posedness of Eq. (2.7), we first consider φ ∈ L ∞ ([0 , T ]; U )and take e A t ( u ) := A ( u ) + ¯ F ( u ) + G ( u ) φ t . Since F ( x, y ) is Lipschitz continuous with respect to x and y , one can show that the map¯ F is also Lipschitz by Lemma 3.2. Indeed, there is a constant C > k ¯ F ( u ) − ¯ F ( v ) k H = (cid:13)(cid:13)(cid:13) Z H F ( u, z ) µ u ( dz ) − Z H F ( v, z ) µ v ( dz ) (cid:13)(cid:13)(cid:13) H ≤ (cid:13)(cid:13)(cid:13) Z H F ( u, z ) µ u ( dz ) − E F ( u, Y u,yt ) (cid:13)(cid:13)(cid:13) H + (cid:13)(cid:13)(cid:13) Z H F ( v, z ) µ u ( dz ) − E F ( v, Y v,yt ) (cid:13)(cid:13)(cid:13) H + (cid:13)(cid:13)(cid:13) E F ( u, Y u,yt ) − E F ( v, Y v,yt ) (cid:13)(cid:13)(cid:13) H ≤ C (1 + k x k H + k y k H ) e − ̺t + C (cid:0) k u − v k H + E k Y u,yt − Y v,yt k H (cid:1) ≤ C (1 + k x k H + k y k H ) e − ̺t + C k u − v k H . Taking t ↑ + ∞ yields that ¯ F is Lipschitz continuous. By Hypothesis 2.1, it is easyto check that e A t ( u ) satisfies the local monotonicity, coercivity conditions established bythe third named author and R¨ockner [24, Theorem 5.1.3] since ¯ F is Lipschitz and φ ∈ L ∞ ([0 , T ]; U ). Hence Eq. (2.7) admits a unique solution for any φ ∈ L ∞ ([0 , T ]; U ) satisfying¯ X φ ∈ C ([0 , T ]; H ) ∩ L γ ([0 , T ]; V ). Step 2 : For any φ ∈ L ([0 , T ]; U ), one can choose a sequence φ n ∈ L ∞ ([0 , T ]; U ) suchthat φ n strongly converge to φ in L ([0 , T ]; U ) as n → ∞ . Let ¯ X φ n denote a unique solutionto Eq. (2.7) with φ n ∈ L ∞ ([0 , T ]; U ). According to ( A2 ), it follows that there are some10onstants C >
0, for any n, m ∈ N we have ddt k ¯ X φ n t − ¯ X φ m t k H = 2 V ∗ h A ( ¯ X φ n t ) − A ( ¯ X φ m t ) , ¯ X φ n t − ¯ X φ m t i V + 2 h ¯ F ( ¯ X φ n t ) − ¯ F ( ¯ X φ m t ) , ¯ X φ n t − ¯ X φ m t i H +2 h G ( ¯ X φ n t ) φ nt − G ( ¯ X φ m t ) φ mt , ¯ X φ n t − ¯ X φ m t i H ≤ − θ k ¯ X φ n t − ¯ X φ m t k γ V + ( C + ρ ( ¯ X φ m t )) k ¯ X φ n t − ¯ X φ m t k H + k φ nt k U k ¯ X φ n t − ¯ X φ m t k H +2 h G ( ¯ X φ m t )( φ nt − φ mt ) , ¯ X φ n t − ¯ X φ m t i H ≤ − θ k ¯ X φ n t − ¯ X φ m t k γ V + k φ nt − φ mt k U +( C + ρ ( ¯ X φ m t ) + k φ nt k U + k G ( ¯ X φ m t ) k L ( U,H ) ) k ¯ X φ n t − ¯ X φ m t k H . (3.5)Gronwall’ lemma yields that k ¯ X φ n t − ¯ X φ m t k H + θ Z t k ¯ X φ n s − ¯ X φ m s k γ V ds ≤ exp n Z T (cid:16) C + ρ ( ¯ X φ m t ) + k φ nt k U + k G ( ¯ X φ m t ) k L ( U,H ) (cid:17) dt o Z T k φ nt − φ mt k U dt. (3.6)Following the similar calculations as in (3.5) that ddt k ¯ X φ m t k H ≤ V ∗ h A ( ¯ X φ m t ) , ¯ X φ m t i V + 2 h ¯ F ( ¯ X φ m t ) + G ( ¯ X φ m t ) φ mt , ¯ X φ m t i H ≤ − θ k ¯ X φ m t k γ V + C (1 + k φ mt k U ) k ¯ X φ m t k H + C. Using Gronwall’ lemma and supposing that φ m ∈ S M leads tosup t ∈ [0 ,T ] k ¯ X φ m t k H + θ Z T k ¯ X φ m t k γ V dt ≤ C exp n Z T (cid:16) k φ mt k U (cid:17) dt o ( k x k H + T ) ≤ C T,M (1 + k x k H ) , (3.7)where constant C T,M only depends on
T, M . Owing to ( A3 ), it follows that Z T k G ( ¯ X φ m t ) k L ( U,H ) dt ≤ C Z T (1 + k ¯ X φ m t k H ) dt ≤ C T,M (1 + k x k H ) . (3.8)Substituting (3.7) and (3.8) into (3.6) and letting n → ∞ , we obtain that { ¯ X φ n } n ≥ is aCauchy net in C ([0 , T ]; H ) ∩ L γ ([0 , T ]; V ), then the limit is denoted by ¯ X φ . Followingthe standard monotonicity argument (see e.g. [40, Theorem 30.A]) implies that ¯ X φ is thesolution to Eq. (2.7) associated with φ , the uniqueness is a direct consequence via ( A2 ) andGronwall’s lemma, and the estimate (3.4) can be concluded by (3.7). Hence, we completethe proof of Lemma 3.3. (cid:3) We now introduce the following stochastic control problem associated with Eq. (2.1), dX ǫ,α,φ ǫ t = (cid:2) A ( X ǫ,α,φ ǫ t ) + F ( X ǫ,α,φ ǫ t , Y ǫ,α,φ ǫ t ) + G ( X ǫ,α,φ ǫ t ) φ ǫt (cid:3) dt + √ ǫG ( X ǫ,α,φ ǫ t ) dW t ,dY ǫ,α,φ ǫ t = 1 α F ( X ǫ,α,φ ǫ t , Y ǫ,α,φ ǫ t ) dt + 1 √ αǫ G ( X ǫ,α,φ ǫ t , Y ǫ,α,φ ǫ t ) φ ǫt dt + 1 √ α G ( X ǫ,α,φ ǫ t , Y ǫ,α,φ ǫ t ) dW t ,X ǫ,α,φ ǫ = x, Y ǫ,α,φ ǫ = y, (3.9)11here { φ ǫ } ǫ> ⊂ A M for some M < ∞ . Let us defineΦ( t ) := exp n − √ ǫ Z t h φ ǫs , dW s i U − ǫ Z t k φ ǫs k U ds o . It is obvious that the so-called Novikov’s condition are satisfied with Φ( t ) due to φ ǫ ∈ A M ,therefore Φ( t ) is a martingale. Hence, one can consider the following weighted probabilitymeasure on (Ω , F , P ) d b P := Φ( T ) d P . Thanks to the Girsanov’s theorem, the process W ǫt := W t + 1 √ ǫ Z t φ ǫs ds, t ∈ [0 , T ] (3.10)is a cylindrical Wiener process with respect to the stochastic basis (Ω , F , F t , b P ). Accordingto the uniqueness of solution to Eq. (2.1) and Yamada-Watanabe theorem, it is easy to seethat X ǫ,α,φ ǫ · := G ǫ (cid:16) W · + 1 √ ǫ Z · φ ǫs ds (cid:17) (3.11)is a part of solution ( X ǫ,α,φ ǫ , Y ǫ,α,φ ǫ ) to Eq. (3.9) with W ǫ · instead of W · on (Ω , F , F t , b P ).Since the probability measure P and b P are mutually absolutely continuous, this implies that(3.11) is also a part of solution ( X ǫ,α,φ ǫ , Y ǫ,α,φ ǫ ) on (Ω , F , F t , P ).Some energy estimates of the solutions ( X ǫ,α,φ ǫ , Y ǫ,α,φ ǫ ) to the controlled equation (3.9)are derived as follows. Lemma 3.4
For each φ ǫ ∈ A M , M ∈ (0 , + ∞ ) , there are some constants C > such thatfor any ǫ, α ∈ (0 , , E h sup t ∈ [0 ,T ] k X ǫ,α,φ ǫ t k H + 2 θ Z T k X ǫ,α,φ ǫ t k γ V dt i ≤ C (1 + k x k H + k y k H ) (3.12) and E Z T k Y ǫ,α,φ ǫ t k H dt ≤ C (1 + k x k H + k y k H ) . (3.13) Proof
Applying Itˆo’s formula for k Y ǫ,α,φ ǫ t k H gives that k Y ǫ,α,φ ǫ t k H = k y k H + 2 α Z t V ∗ h F ( X ǫ,α,φ ǫ s , Y ǫ,α,φ ǫ s ) , Y ǫ,α,φ ǫ s i V ds + 1 α Z t k G ( X ǫ,α,φ ǫ s , Y ǫ,α,φ ǫ s ) k L ( U,H ) ds + 2 √ αǫ Z t h G ( X ǫ,α,φ ǫ s , Y ǫ,α,φ ǫ s ) φ ǫs , Y ǫ,α,φ ǫ s i H ds + 2 √ α Z t h G ( X ǫ,α,φ ǫ s , Y ǫ,α,φ ǫ s ) dW s , Y ǫ,α,φ ǫ s i H ds. Taking expectation and differentiating with respect to t leads to ddt E k Y ǫ,α,φ ǫ t k H = 2 α E h V ∗ h F ( X ǫ,α,φ ǫ t , Y ǫ,α,φ ǫ t ) , Y ǫ,α,φ ǫ t i V i + 1 α E h k G ( X ǫ,α,φ ǫ t , Y ǫ,α,φ ǫ t ) k L ( U,H ) i + 2 √ αǫ E h h G ( X ǫ,α,φ ǫ t , Y ǫ,α,φ ǫ t ) φ ǫt , Y ǫ,α,φ ǫ t i H i . (3.14)12ollowing the same calculations as in the proof of [24, Lemma 4.3.8] that by Hypothesis 2.2,there is a constant η ∈ (0 , θ ),2 V ∗ h F ( u, v ) , v i V ≤ − η k v k H + C (1 + k u k H ) . Hence, the first term of the right hand side of (3.14) is controlled by1 α E h V ∗ h F ( X ǫ,α,φ ǫ t , Y ǫ,α,φ ǫ t ) , Y ǫ,α,φ ǫ t i V i ≤ − ηα E k Y ǫ,α,φ ǫ t k H + Cα (1 + E k X ǫ,α,φ ǫ t k H ) . (3.15)The last term of the right hand side of (3.14) can be estimated by2 √ αǫ E h h G ( X ǫ,α,φ ǫ t , Y ǫ,α,φ ǫ t ) φ ǫt , Y ǫ,α,φ ǫ t i H i ≤ √ αǫ E h k G ( X ǫ,α,φ ǫ t , Y ǫ,α,φ ǫ t ) k L ( U,H ) k φ ǫt k U k Y ǫ,α,φ ǫ t k H i ≤ C √ αǫ E h (1 + k X ǫ,α,φ ǫ t k H ) k φ ǫt k U k Y ǫ,α,φ ǫ t k H i ≤ Cǫ E h (1 + k X ǫ,α,φ ǫ t k H ) k φ ǫt k U i + ε α E k Y ǫ,α,φ ǫ t k H , (3.16)where ε ∈ (0 , η ), we used the condition (2.3) in the second step and Young’s inequality inthe last one.Substituting (3 .
15) and (3 .
16) into (3 .
14) and following from (2.3) again yields that ddt E k Y ǫ,α,φ ǫ t k H ≤ − C η α E k Y ǫ,α,φ ǫ t k H + Cα (1 + E k X ǫ,α,φ ǫ t k H ) + Cǫ E h (1 + k X ǫ,α,φ ǫ t k H ) k φ ǫt k U i . The comparison theorem implies E k Y ǫ,α,φ ǫ t k H ≤ e − Cηα t k y k H + Cα Z t e − Cηα ( t − s ) (1 + E k X ǫ,α,φ ǫ s k H ) ds + Cǫ Z t e − Cηα ( t − s ) E h (1 + k X ǫ,α,φ ǫ s k H ) k φ ǫs k U i ds. (3.17)Integrating (3.17) with respect to t from 0 to T and using Fubini’s theorem leads to E h Z T k Y ǫ,α,φ ǫ t k H dt i ≤ k y k H Z T e − Cηα t dt + Cα Z T Z t e − Cηα ( t − s ) (1 + E k X ǫ,α,φ ǫ s k H ) dsdt + Cǫ E h sup t ∈ [0 ,T ] (1 + k X ǫ,α,φ ǫ t k H ) Z T Z t e − Cηα ( t − s ) k φ ǫs k U dsdt i ≤ αC η k y k H + C η E h Z T (1 + k X ǫ,α,φ ǫ t k H ) dt i + C η (cid:16) αǫ (cid:17) E h sup t ∈ [0 ,T ] (1 + k X ǫ,α,φ ǫ t k H ) Z T k φ ǫt k U dt i ≤ C α,η,T (1 + k y k H ) + C η E Z T k X ǫ,α,φ ǫ t k H dt + C η,M (cid:16) αǫ (cid:17)h E (cid:16) sup t ∈ [0 ,T ] k X ǫ,α,φ ǫ t k H (cid:17)i . (3.18)13ow we aim to estimate k X ǫ,α,φ ǫ t k H . First, according to Itˆo’s formula k X ǫ,α,φ ǫ t k H = k x k H + 2 Z t V ∗ h A ( X ǫ,α,φ ǫ s ) , X ǫ,α,φ ǫ s i V ds + ǫ Z t k G ( X ǫ,α,φ ǫ s ) k L ( U,H ) ds +2 Z t h h F ( X ǫ,α,φ ǫ s , Y ǫ,α,φ ǫ s ) + G ( X ǫ,α,φ ǫ s ) φ ǫs , X ǫ,α,φ ǫ s i H i ds +2 √ ǫ Z t h G ( X ǫ,α,φ ǫ s ) dW s , X ǫ,α,φ ǫ s i H . Following from Remark 2.1 and ( A3 ) that E h sup s ∈ [0 ,t ] k X ǫ,α,φ ǫ s k H i + θ E Z t k X ǫ,α,φ ǫ s k γ V ds ≤ k x k H + CT + C E Z t k X ǫ,α,φ ǫ s k H ds + C E Z t k Y ǫ,α,φ ǫ s k H ds +2 E Z t |h G ( X ǫ,α,φ ǫ s ) φ ǫs , X ǫ,α,φ ǫ s i H | ds + 2 √ ǫ E h sup s ∈ [0 ,t ] (cid:12)(cid:12)(cid:12) Z s h G ( X ǫ,α,φ ǫ r ) dW r , X ǫ,α,φ ǫ r i H (cid:12)(cid:12)(cid:12)i . (3.19)Using Cauchy-Schwarz’s inequality, H¨older’s inequality and Young’s inequality that2 E Z t |h G ( X ǫ,α,φ ǫ s ) φ ǫs , X ǫ,α,φ ǫ s i H | ds ≤ E h sup s ∈ [0 ,t ] k X ǫ,α,φ ǫ s k H Z t k G ( X ǫ,α,φ ǫ s ) k L ( U,H ) k φ ǫs k U ds i ≤ E h sup s ∈ [0 ,t ] k X ǫ,α,φ ǫ s k H i + 4 E (cid:16) Z t k G ( X ǫ,α,φ ǫ s ) k L ( U,H ) k φ ǫs k U ds (cid:17) ≤ E h sup s ∈ [0 ,t ] k X ǫ,α,φ ǫ s k H i + 4 E h(cid:16) Z t k G ( X ǫ,α,φ ǫ s ) k L ( U,H ) ds (cid:17)(cid:16) Z T k φ ǫs k U ds (cid:17)i ≤ E h sup s ∈ [0 ,t ] k X ǫ,α,φ ǫ s k H i + C M,T + C M E Z t k X ǫ,α,φ ǫ s k H ds, (3.20)where the last step is due to φ ǫ ∈ A M . From Burkholder-Davis-Gundy’s inequality, the lastterm of right hand side of (3.19) is estimated by the following2 √ ǫ E h sup s ∈ [0 ,t ] (cid:12)(cid:12)(cid:12) Z s h G ( X ǫ,α,φ ǫ r ) dW r , X ǫ,α,φ ǫ r i H (cid:12)(cid:12)(cid:12)i ≤ √ ǫ E h Z t k G ( X ǫ,α,φ ǫ s ) k L ( U,H ) k X ǫ,α,φ ǫ s k H ds i ≤ √ ǫ E h sup s ∈ [0 ,t ] k X ǫ,α,φ ǫ s k H Z t k G ( X ǫ,α,φ ǫ s ) k L ( U,H ) i ≤ E h sup s ∈ [0 ,t ] k X ǫ,α,φ ǫ s k H i + C ǫ E Z t k X ǫ,α,φ ǫ s k H ds + C ǫ,T . (3.21)14ubstituting (3.20) and (3.21) into (3.19) and recalling (3.18), we infer that E h sup s ∈ [0 ,t ] k X ǫ,α,φ ǫ s k H i + 2 θ E Z t k X ǫ,α,φ ǫ s k γ V ds ≤ k x k H + C M,ǫ,T + C M,ǫ,T E Z t k X ǫ,α,φ ǫ s k H ds + C E Z t k Y ǫ,α,φ ǫ s k H ds ≤ k x k H + C α,η,T (1 + k y k H ) + C M,ǫ,T + C M,ǫ,T,η E Z t k X ǫ,α,φ ǫ s k H ds + C M,η (cid:16) αǫ (cid:17)h E (cid:16) sup t ∈ [0 ,T ] k X ǫ,α,φ ǫ t k H (cid:17)i . Owing to the additional condition (2.8), one can take αǫ < C η,M such that E h sup s ∈ [0 ,t ] k X ǫ,α,φ ǫ s k H i + 2 θ E Z t k X ǫ,α,φ ǫ s k γ V ds ≤ C M,ǫ,T,η h k x k H + k y k H + E Z t k X ǫ,α,φ ǫ s k H ds i . Consequently, the Gronwall’s lemma yields that E h sup s ∈ [0 ,t ] k X ǫ,α,φ ǫ s k H i + 2 θ E Z t k X ǫ,α,φ ǫ s k γ V ds ≤ C M,ǫ,T,η h k x k H + k y k H i , (3.22)which gives the estimate (3.12).It is easy to get the estimate (3.13) by substituting (3.22) into (3.18). The proof of thislemma is completed. (cid:3) Now we would like to formulate a technical lemma, which investigates time incrementsof the solution to the controlled equation (3.9). In order to state this lemma, we first definethe following stopping time τ ǫN := inf n t ∈ [0 , T ] : k X ǫ,α,φ ǫ t k H > N o , for any ǫ, N > Lemma 3.5
For each x ∈ H , y ∈ H , T > and ǫ, δ ∈ (0 , are small enough constants,there are some constants C N > depending on N such that E h Z T ∧ τ ǫN k X ǫ,α,φ ǫ t − X ǫ,α,φ ǫ t ( δ ) k H dt i ≤ C N (1 + k x k H + k y k H ) δ , here t ( δ ) := [ tδ ] δ and [ s ] is the largest integer smaller than s .Proof It is easy to see that E h Z T ∧ τ ǫN k X ǫ,α,φ ǫ t − X ǫ,α,φ ǫ t ( δ ) k H dt i ≤ E h Z δ k X ǫ,α,φ ǫ t − x k H { t ≤ τ ǫN } dt i + E h Z Tδ k X ǫ,α,φ ǫ t − X ǫ,α,φ ǫ t ( δ ) k H { t ≤ τ ǫN } dt i ≤ C N (1 + k x k H ) δ + 2 E h Z Tδ k X ǫ,α,φ ǫ t − X ǫ,α,φ ǫ t − δ k H { t ≤ τ ǫN } dt i +2 E h Z Tδ k X ǫ,α,φ ǫ t ( δ ) − X ǫ,α,φ ǫ t − δ k H { t ≤ τ ǫN } dt i . (3.23)15e first work on the second term of right hand side of (3.23). Applying Itˆo’s formula yieldsthat k X ǫ,α,φ ǫ t − X ǫ,α,φ ǫ t − δ k H ≤ Z tt − δ V ∗ h A ( X ǫ,α,φ ǫ s ) , X ǫ,α,φ ǫ s − X ǫ,α,φ ǫ t − δ i V ds +2 Z tt − δ h F ( X ǫ,α,φ ǫ s , Y ǫ,α,φ ǫ s ) , X ǫ,α,φ ǫ s − X ǫ,α,φ ǫ t − δ i H ds +2 Z tt − δ h G ( X ǫ,α,φ ǫ s ) φ ǫs , X ǫ,α,φ ǫ s − X ǫ,α,φ ǫ t − δ i H ds + ǫ Z tt − δ k G ( X ǫ,α,φ ǫ s ) k L ( U,H ) ds +2 √ ǫ Z tt − δ h G ( X ǫ,α,φ ǫ s ) dW s , X ǫ,α,φ ǫ s − X ǫ,α,φ ǫ t − δ i H ds =: X i =1 K i ( t ) . (3.24)Let us now estimate E h R Tδ | K i ( t ) | { t ≤ τ ǫN } dt i , i = 1 , , ...,
5, respectively. According to( A3 ) and H¨older’s inequality, there is a constant C N,T > E h Z Tδ | K ( t ) | { t ≤ τ ǫN } dt i ≤ C E h Z Tδ Z tt − δ k A ( X ǫ,α,φ ǫ s ) k V ∗ k X ǫ,α,φ ǫ s − X ǫ,α,φ ǫ t − δ k V { t ≤ τ ǫN } dsdt i ≤ C h E (cid:16) Z Tδ Z tt − δ k A ( X ǫ,α,φ ǫ s ) k γ γ − V ∗ { t ≤ τ ǫN } dsdt (cid:17)i γ − γ × h E (cid:16) Z Tδ Z tt − δ k X ǫ,α,φ ǫ s − X ǫ,α,φ ǫ t − δ k γ V { t ≤ τ ǫN } dsdt (cid:17)i γ ≤ C h δ E (cid:16) Z T (1 + k X ǫ,α,φ ǫ t k γ V )(1 + k X ǫ,α,φ ǫ t k βH ) { t ≤ τ ǫN } dt (cid:17)i γ − γ h δ E Z T k X ǫ,α,φ ǫ t k γ V dt i γ ≤ C N,T δ (1 + k x k H + k y k H ) , (3.25)where we applied the definition of τ ǫN in the last step, and the third inequality is owing to E (cid:16) Z Tδ Z tt − δ k A ( X ǫ,α,φ ǫ s ) k γ γ − V ∗ { t ≤ τ ǫN } dsdt (cid:17) = E h Z δ k A ( X ǫ,α,φ ǫ s ) k γ γ − V ∗ (cid:16) Z s + δδ { t ≤ τ ǫN } dt (cid:17) ds + Z T − δδ k A ( X ǫ,α,φ ǫ s ) k γ γ − V ∗ (cid:16) Z s + δs { t ≤ τ ǫN } dt (cid:17) ds i + E h Z TT − δ k A ( X ǫ,α,φ ǫ s ) k γ γ − V ∗ (cid:16) Z Ts { t ≤ τ ǫN } dt (cid:17) ds i ≤ δ E h Z T k A ( X ǫ,α,φ ǫ s ) k γ γ − V ∗ { s ≤ τ ǫN } ds i .
16y Remark 2.1, (3.2), (3.3) and H¨older’s inequality, it follows that E h Z Tδ | K ( t ) | { t ≤ τ ǫN } dt i ≤ C h E (cid:16) Z Tδ Z tt − δ k F ( X ǫ,α,φ ǫ s , Y ǫ,α,φ ǫ s ) k H { t ≤ τ ǫN } dsdt (cid:17)i × h E (cid:16) Z Tδ Z tt − δ k X ǫ,α,φ ǫ s − X ǫ,α,φ ǫ t − δ k H { t ≤ τ ǫN } dsdt (cid:17)i ≤ C h δ E (cid:16) Z T (1 + k X ǫ,α,φ ǫ s k H + k Y ǫ,α,φ ǫ s k H ) ds (cid:17)i h δ E (cid:16) Z T k X ǫ,α,φ ǫ s k H ds (cid:17)i ≤ C T δ (1 + k x k H + k y k H ) . (3.26)From the condition ( A3 ), the term E h R Tδ | K ( t ) | { t ≤ τ ǫN } dt i can be controlled by E h Z Tδ | K ( t ) | { t ≤ τ ǫN } dt i ≤ C h E (cid:16) Z Tδ Z tt − δ k G ( X ǫ,α,φ ǫ s ) k L ( U,H ) k φ ǫs k U { t ≤ τ ǫN } dsdt (cid:17)i × h E (cid:16) Z Tδ Z tt − δ k X ǫ,α,φ ǫ s − X ǫ,α,φ ǫ t − δ k H { t ≤ τ ǫN } dsdt (cid:17)i ≤ C h δ E (cid:16) Z T (1 + k X ǫ,α,φ ǫ t k H ) k φ ǫt k U { t ≤ τ ǫN } dt (cid:17)i h δ E (cid:16) Z T k X ǫ,α,φ ǫ s k H ds (cid:17)i ≤ C N,T δ (1 + k x k H + k y k H ) . (3.27)Following the similar calculations, the term E h R Tδ | K ( t ) | { t ≤ τ ǫN } dt i will be estimated by E h Z Tδ | K ( t ) | { t ≤ τ ǫN } dt i ≤ C T δǫ E h t ∈ [0 ,T ] k X ǫ,α,φ ǫ t k H i ≤ C T δ (1 + k x k H + k y k H ) . (3.28)Applying Burkholder-Davis-Gundy’s inequality and (3.12), we indicate that E h Z Tδ | K ( t ) | { t ≤ τ ǫN } dt i ≤ C Z Tδ h E (cid:16) Z tt − δ k G ( X ǫ,α,φ ǫ s ) k L ( U,H ) k X ǫ,α,φ ǫ s − X ǫ,α,φ ǫ t − δ k H { t ≤ τ ǫN } ds (cid:17) i dt ≤ C T h E (cid:16) Z Tδ Z tt − δ (1 + k X ǫ,α,φ ǫ s k H ) k X ǫ,α,φ ǫ s − X ǫ,α,φ ǫ t − δ k H { t ≤ τ ǫN } dsdt (cid:17)i ≤ C N,T δ h E (cid:16) t ∈ [0 ,T ] k X ǫ,α,φ ǫ t k H (cid:17)i ≤ C N,T δ (1 + k x k H + k y k H ) . (3.29)Substituting (3 . − (3 .
29) into (3 . E h Z Tδ k X ǫ,α,φ ǫ t − X ǫ,α,φ ǫ t − δ k H { t ≤ τ ǫN } dt i ≤ C N,T δ (1 + k x k H + k y k H ) . E h Z Tδ k X ǫ,α,φ ǫ t ( δ ) − X ǫ,α,φ ǫ t − δ k H { t ≤ τ ǫN } dt i ≤ C N,T δ (1 + k x k H + k y k H ) , which combining with (3 .
23) implies the assertion. The proof is completed. (cid:3)
In this section, we aim to verify the main results formulated in Theorem 2.1. We firstconsider an auxiliary equation associated with the fast component of Eq. (2.1), which willhelp us to verify the convergence of the stochastic controlled equation (3.9) to the solutionof skeleton equation (2.7). After the above-mentioned preparation, we are in a position toinvestigate the LDP of { X ǫ,α : ǫ > } to the two-time-scale Eq. (2.1) on C ([0 , T ]; H ) ∩ L γ ([0 , T ]; V ). Since we want to use the approach of time discretization developed by Khasminskii [18],we establish the following auxiliary process b Y ǫ,αt ∈ H and separate the time interval [0 , T ]into some subintervals of size δ > α and will be chosen in the next subsection.Consider d b Y ǫ,αt = 1 α F ( X ǫ,α,φ ǫ t ( δ ) , b Y ǫ,αt ) dt + 1 √ α G ( X ǫ,α,φ ǫ t ( δ ) , b Y ǫ,αt ) dW t , b Y ǫ,α = y. (4.1)It is easy to see that for each k ∈ N and t ∈ [ kδ, ( k + 1) δ ∧ T ], b Y ǫ,αt = b Y ǫ,αkδ + 1 α Z tkδ F ( X ǫ,α,φ ǫ kδ , b Y ǫ,αs ) ds + 1 √ α Z tkδ G ( X ǫ,α,φ ǫ kδ , b Y ǫ,αs ) dW s . (4.2)Following the almost same calculations as in the proof of Lemma 3.4, one can easily getthe energy estimate for b Y ǫ,αt as follows, we omit the details to save space. Lemma 4.1
For any initial values x ∈ H , y ∈ H and ǫ, α ∈ (0 , , there is a constant C > such that sup t ∈ [0 ,T ] E k b Y ǫ,αt k H ≤ C (1 + k x k H + k y k H ) . (4.3)Now we would like to prove an important lemma characterizing the difference between b Y ǫ,αt and Y ǫ,α,φ ǫ t . Lemma 4.2
For any initial values x ∈ H , y ∈ H and ǫ, α ∈ (0 , , there is a constant C N > such that E h Z T ∧ τ ǫN k Y ǫ,α,φ ǫ t − b Y ǫ,αt k H dt i ≤ C N (1 + k x k H + k y k H ) h(cid:16) αǫ (cid:17) + δ i . Proof
Letting Z t := Y ǫ,α,φ ǫ t − b Y ǫ,αt , which fulfills the following SPDE dZ t = 1 α h F ( X ǫ,α,φ ǫ t , Y ǫ,α,φ ǫ t ) − F ( X ǫ,α,φ ǫ t ( δ ) , b Y ǫ,αt ) i dt + 1 √ αǫ G ( X ǫ,α,φ ǫ t , Y ǫ,α,φ ǫ t ) φ ǫt dt + 1 √ α h G ( X ǫ,α,φ ǫ t , Y ǫ,α,φ ǫ t ) − G ( X ǫ,α,φ ǫ t ( δ ) , b Y ǫ,αt ) i dW t , Z = 0 . k Z t ∧ τ ǫN k H yields that k Z t ∧ τ ǫN k H = 2 α Z t h V ∗ h F ( X ǫ,α,φ ǫ s , Y ǫ,α,φ ǫ s ) − F ( X ǫ,α,φ ǫ s ( δ ) , b Y ǫ,αs ) , Z s i V { s ≤ τ ǫN } i ds + 2 √ αǫ Z t h h G ( X ǫ,α,φ ǫ s , Y ǫ,α,φ ǫ s ) φ ǫs , Z s i H { s ≤ τ ǫN } i ds + 1 α Z t h k G ( X ǫ,α,φ ǫ s , Y ǫ,α,φ ǫ s ) − G ( X ǫ,α,φ ǫ s ( δ ) , b Y ǫ,αs ) k L ( U,H ) { s ≤ τ ǫN } i ds + 2 √ α Z t { s ≤ τ ǫN } (cid:10)(cid:0) G ( X ǫ,α,φ ǫ s , Y ǫ,α,φ ǫ s ) − G ( X ǫ,α,φ ǫ s ( δ ) , b Y ǫ,αs ) (cid:1) dW s , Z s (cid:11) H . Taking expectation and differentiating with respect to t , then it follows that ddt E k Z t ∧ τ ǫN k H = 2 α E h V ∗ h F ( X ǫ,α,φ ǫ t , Y ǫ,α,φ ǫ t ) − F ( X ǫ,α,φ ǫ t ( δ ) , b Y ǫ,αt ) , Z t i V { t ≤ τ ǫN } i + 2 √ αǫ E h h G ( X ǫ,α,φ ǫ t , Y ǫ,α,φ ǫ t ) φ ǫt , Z t i H { t ≤ τ ǫN } i + 1 α E h k G ( X ǫ,α,φ ǫ t , Y ǫ,α,φ ǫ t ) − G ( X ǫ,α,φ ǫ t ( δ ) , b Y ǫ,αt ) k L ( U,H ) { t ≤ τ ǫN } i =: X i =1 I i ( t ) . (4.4)Let us estimate the terms I i ( t ), i = 1 , ,
3, respectively. Taking the condition ( H2 ) intoaccount, we have I ( t ) = 2 α E h V ∗ h F ( X ǫ,α,φ ǫ t , Y ǫ,α,φ ǫ t ) − F ( X ǫ,α,φ ǫ t ( δ ) , Y ǫ,α,φ ǫ t ) , Z t i V { t ≤ τ ǫN } i + 2 α E h V ∗ h F ( X ǫ,α,φ ǫ t ( δ ) , Y ǫ,α,φ ǫ t ) − F ( X ǫ,α,φ ǫ t ( δ ) , b Y ǫ,αt ) , Z t i V { t ≤ τ ǫN } i ≤ Cα E h k X ǫ,α,φ ǫ t − X ǫ,α,φ ǫ t ( δ ) k H k Z t k H { t ≤ τ ǫN } i − κα E k Z t ∧ τ ǫN k H ≤ − κ − ε α E k Z t ∧ τ ǫN k H + Cα E h k X ǫ,α,φ ǫ t − X ǫ,α,φ ǫ t ( δ ) k H { t ≤ τ ǫN } i , (4.5)where we used Young’s inequality in the last step with a small enough constant ε > I ( t ) ≤ √ αǫ E h k G ( X ǫ,α,φ ǫ t , Y ǫ,α,φ ǫ t ) k L ( U,H ) k φ ǫt k U k Z t k H { t ≤ τ ǫN } i ≤ ε α E k Z t ∧ τ ǫN k H + Cǫ E h (1 + k X ǫ,α,φ ǫ t k H ) k φ ǫt k U i Following the similar calculations as in (4.5) that I ( t ) ≤ L G α E k Z t ∧ τ ǫN k H + Cα E h k X ǫ,α,φ ǫ t − X ǫ,α,φ ǫ t ( δ ) k H { t ≤ τ ǫN } i . (4.6)Combining (4.5)-(4.6) and substituting them into (4.4) leads to ddt E k Z t ∧ τ ǫN k H ≤ − κ − L G − ε α E k Z t ∧ τ ǫN k H + Cǫ E h (1 + k X ǫ,α,φ ǫ t k H ) k φ ǫt k U i + Cα E h k X ǫ,α,φ ǫ t − X ǫ,α,φ ǫ t ( δ ) k H { t ≤ τ ǫN } i .
19y the comparison theorem we have E k Z t ∧ τ ǫN k H ≤ Cǫ Z t e − ηα ( t − s ) E h (1 + k X ǫ,α,φ ǫ s k H ) k φ ǫs k U i ds + Cα Z t e − ηα ( t − s ) E h k X ǫ,α,φ ǫ s − X ǫ,α,φ ǫ s ( δ ) k H { s ≤ τ ǫN } i ds, here we denote η := 2 κ − L G − ε >
0. Hence, according to Fubini’s theorem, one canconclude that E Z T ∧ τ ǫN k Z t k H dt = Z T E k Z t ∧ τ ǫN k H dt ≤ Cǫ Z T Z t e − ηα ( t − s ) E h (1 + k X ǫ,α,φ ǫ s k H ) k φ ǫs k U i dsdt + Cα Z T Z t e − ηα ( t − s ) E h k X ǫ,α,φ ǫ s − X ǫ,α,φ ǫ s ( δ ) k H { s ≤ τ ǫN } i dsdt ≤ Cǫ E h Z T (1 + k X ǫ,α,φ ǫ s k H ) k φ ǫs k U (cid:16) Z Ts e − ηα ( t − s ) dt (cid:17) ds i + Cα E h Z T { s ≤ τ ǫN } k X ǫ,α,φ ǫ s − X ǫ,α,φ ǫ s ( δ ) k H (cid:16) Z Ts e − ηα ( t − s ) dt (cid:17) ds i ≤ Cη (cid:16) αǫ (cid:17) E h sup t ∈ [0 ,T ] (1 + k X ǫ,α,φ ǫ t k H ) Z T k φ ǫt k U dt i + Cη E h Z T ∧ τ ǫN k X ǫ,α,φ ǫ t − X ǫ,α,φ ǫ t ( δ ) k H dt i ≤ C N,T,M (1 + k x k H + k y k H ) h(cid:16) αǫ (cid:17) + δ i , where the last inequality is owing to Lemma 3.4 and Lemma 3.5, which completes theassertion. (cid:3) In this subsection, our aim is to prove the convergence in distribution of process X ǫ,α,φ ǫ t defined in Eq. (3.9) to the solution ¯ X φt of skeleton equation (2.7), which implies the Condi-tion (A) (i).Repeating the almost same arguments as in the proof of Lemma 3.5, one can easilyconclude the following lemma.
Lemma 4.3
For each x ∈ H and δ > small enough, there is a constant C > such that sup φ ∈S M Z T k ¯ X φt − ¯ X φt ( δ ) k H dt ≤ C (1 + k x k H ) δ . Now we define the following stopping time e τ ǫN := inf n t ∈ [0 , T ] : k X ǫ,α,φ ǫ t k H + k ¯ X φt k H + Z t k X ǫ,α,φ ǫ s k γ V ds + Z t k ¯ X φs k γ V ds > N o . heorem 4.1 Assume that the conditions given in Theorem 2.1 hold. Let { φ ǫ : ǫ > } ⊂A M for some M < ∞ . If φ ǫ converge to φ in distribution as S M -valued random elements,then G ǫ (cid:18) W · + 1 √ ǫ Z · φ ǫs ds (cid:19) → G (cid:18)Z · φ s ds (cid:19) in distribution as ǫ → .Proof We will separate the proof into four steps to prove the convergence of solutionof Eq. (2.1) to the solution of Eq. (2.7) in probability, which implies the convergence indistribution as ǫ → Step 1 : Denote e Z ǫt := X ǫ,α,φ ǫ t − ¯ X φt , which satisfies the following SPDE d e Z ǫt = (cid:2) A ( X ǫ,α,φ ǫ t ) − A ( ¯ X φt ) (cid:3) dt + + (cid:2) F ( X ǫ,α,φ ǫ t , Y ǫ,α,φ ǫ t ) − ¯ F ( ¯ X φt ) (cid:3) dt + (cid:2) G ( X ǫ,α,φ ǫ t ) φ ǫt − G ( ¯ X φt ) φ t (cid:3) dt + √ ǫG ( X ǫ,α,φ ǫ t ) dW t , e Z ǫ =0 . Applying Itˆo’s formula to k e Z ǫt k H we obtain k e Z ǫt k H = 2 Z t V ∗ h A ( X ǫ,α,φ ǫ s ) − A ( ¯ X φs ) , e Z ǫs i V ds + 2 Z t h F ( X ǫ,α,φ ǫ s , Y ǫ,α,φ ǫ s ) − ¯ F ( ¯ X φs ) , e Z ǫs i H ds +2 Z t h G ( X ǫ,α,φ ǫ s ) φ ǫs − G ( ¯ X φs ) φ s , e Z ǫs i H ds + ǫ Z t k G ( X ǫ,α,φ ǫ s ) k L ( U,H ) ds +2 √ ǫ Z t h e Z ǫs , G ( X ǫ,α,φ ǫ s ) dW s i H = 2 Z t V ∗ h A ( X ǫ,α,φ ǫ s ) − A ( ¯ X φs ) , e Z ǫs i V ds + 2 Z t h ¯ F ( X ǫ,α,φ ǫ s ) − ¯ F ( ¯ X φs ) , e Z ǫs i H ds +2 Z t h F ( X ǫ,α,φ ǫ s , Y ǫ,α,φ ǫ s ) − ¯ F ( X ǫ,α,φ ǫ s ) − F ( X ǫ,α,φ ǫ s ( δ ) , b Y ǫ,αs ) + ¯ F ( X ǫ,α,φ ǫ s ( δ ) ) , e Z ǫs i H ds +2 Z t h F ( X ǫ,α,φ ǫ s ( δ ) , b Y ǫ,αs ) − ¯ F ( X ǫ,α,φ ǫ s ( δ ) ) , e Z ǫs − e Z ǫs ( δ ) i H ds +2 Z t h F ( X ǫ,α,φ ǫ s ( δ ) , b Y ǫ,αs ) − ¯ F ( X ǫ,α,φ ǫ s ( δ ) ) , e Z ǫs ( δ ) i H ds +2 Z t h (cid:2) G ( X ǫ,α,φ ǫ s ) − G ( ¯ X φs ) (cid:3) φ ǫs , e Z ǫs i H ds + 2 Z t h G ( ¯ X φs )( φ ǫs − φ s ) , e Z ǫs i H ds + ǫ Z t k G ( X ǫ,α,φ ǫ s ) k L ( U,H ) ds + 2 √ ǫ Z t h e Z ǫs , G ( X ǫ,α,φ ǫ s ) dW s i H =: X i =1 I i ( t ) . (4.7)Taking ( A2 ) and Young’s inequality into account, we have I ( t ) + I ( t ) ≤ Z t V ∗ h A ( X ǫ,α,φ ǫ s ) − A ( ¯ X φs ) , e Z ǫs i V + k G ( X ǫ,α,φ ǫ s ) − G ( ¯ X φs ) k L ( U,H ) ds + Z t k φ ǫs k U k e Z ǫs k H ds ≤ − θ Z t k e Z ǫs k γ V ds + Z t (cid:0) K + ρ ( X ǫ,α,φ ǫ s ) (cid:1) k e Z ǫs k H ds + Z t k φ ǫs k U k e Z ǫs k H ds. (4.8)21ince ¯ F is Lipschitz continuous from the proof of Lemma 3.3, it is obvious that I ( t ) ≤ Z t k ¯ F ( X ǫ,α,φ ǫ s ) − ¯ F ( ¯ X φs ) k H k e Z ǫs k H ds ≤ C Z t k e Z ǫs k H ds. (4.9)Similarly, by ( A2 ) and Young’s inequality, it leads to I ( t ) ≤ Z t k e Z ǫs k H ds + C Z t (cid:16) k X ǫ,α,φ ǫ s − X ǫ,α,φ ǫ s ( δ ) k H + k Y ǫ,α,φ ǫ s − b Y ǫ,αs k H (cid:17) ds. (4.10)Following from Young’s inequality and H¨older’s inequality yields that I ( t ) ≤ Z t k F ( X ǫ,α,φ ǫ s ( δ ) , b Y ǫ,αs ) − ¯ F ( X ǫ,α,φ ǫ s ( δ ) ) k H k e Z ǫs − e Z ǫs ( δ ) k H ds ≤ h Z t (cid:16) k F ( X ǫ,α,φ ǫ s ( δ ) , b Y ǫ,αs ) k H + k ¯ F ( X ǫ,α,φ ǫ s ( δ ) ) k H (cid:17) ds i × h Z t (cid:16) k X ǫ,α,φ ǫ s − X ǫ,α,φ ǫ s ( δ ) k H + k ¯ X φs − ¯ X φs ( δ ) k H (cid:17) ds i ≤ C h Z t (cid:16) k X ǫ,α,φ ǫ s ( δ ) k H + k b Y ǫ,αs k H (cid:17) ds i × h Z t (cid:16) k X ǫ,α,φ ǫ s − X ǫ,α,φ ǫ s ( δ ) k H + k ¯ X φs − ¯ X φs ( δ ) k H (cid:17) ds i . (4.11)Substituting (4.8)-(4.11) into (4.7) and then rearranging it gives k e Z ǫt k H + θ Z t k e Z ǫs k γ V ds ≤ C Z t (cid:0) ρ ( X ǫ,α,φ ǫ s ) + k φ ǫs k U (cid:1) k e Z ǫs k H ds + Cǫ Z t (cid:0) k ¯ X φs k H (cid:1) ds + C Z t (cid:16) k X ǫ,α,φ ǫ s − X ǫ,α,φ ǫ s ( δ ) k H + k Y ǫ,α,φ ǫ s − b Y ǫ,αs k H (cid:17) ds + C h Z t (cid:16) k X ǫ,α,φ ǫ s ( δ ) k H + k b Y ǫ,αs k H (cid:17) ds i h Z t (cid:16) k X ǫ,α,φ ǫ s − X ǫ,α,φ ǫ s ( δ ) k H + k ¯ X φs − ¯ X φs ( δ ) k H (cid:17) ds i +2 Z t h G ( ¯ X φs )( φ ǫs − φ s ) , e Z ǫs i H ds + 2 √ ǫ Z t h e Z ǫs , G ( X ǫ,α,φ ǫ s ) dW s i H + ǫ Z t k G ( X ǫ,α,φ ǫ s ) k L ( U,H ) ds + 2 Z t h F ( X ǫ,α,φ ǫ s ( δ ) , b Y ǫ,αs ) − ¯ F ( X ǫ,α,φ ǫ s ( δ ) ) , e Z ǫs ( δ ) i H ds. e τ ǫN , it follows thatsup t ∈ [0 ,T ∧ e τ ǫN ] k e Z ǫt k H + θ Z T ∧ e τ ǫN k e Z ǫt k γ V dt ≤ C N n ǫ Z T (cid:0) k ¯ X φt k H (cid:1) dt + Z T ∧ e τ ǫN (cid:16) k X ǫ,α,φ ǫ t − X ǫ,α,φ ǫ t ( δ ) k H + k Y ǫ,α,φ ǫ t − b Y ǫ,αt k H (cid:17) dt + ǫ Z T ∧ e τ ǫN k G ( X ǫ,α,φ ǫ t ) k L ( U,H ) dt + h Z T (cid:16) k X ǫ,α,φ ǫ t ( δ ) k H + k b Y ǫ,αt k H (cid:17) dt i × h Z T ∧ e τ ǫN k X ǫ,α,φ ǫ t − X ǫ,α,φ ǫ t ( δ ) k H dt + Z T k ¯ X φt − ¯ X φt ( δ ) k H dt i + sup t ∈ [0 ,T ∧ e τ ǫN ] (cid:12)(cid:12)(cid:12) Z t h G ( ¯ X φs )( φ ǫs − φ s ) , e Z ǫs i H ds (cid:12)(cid:12)(cid:12) + √ ǫ sup t ∈ [0 ,T ∧ e τ ǫN ] (cid:12)(cid:12)(cid:12) Z t h e Z ǫt , G ( X ǫ,α,φ ǫ t ) dW t i H (cid:12)(cid:12)(cid:12) + sup t ∈ [0 ,T ∧ e τ ǫN ] (cid:12)(cid:12)(cid:12) Z t h F ( X ǫ,α,φ ǫ s ( δ ) , b Y ǫ,αs ) − ¯ F ( X ǫ,α,φ ǫ s ( δ ) ) , e Z ǫs ( δ ) i H ds (cid:12)(cid:12)(cid:12)o × exp (cid:16) Z T k φ ǫt k U dt (cid:17) . Taking expectation for both sides of the above inequality and using Lemma 3.5, 4.1, 4.2 and4.3 yields E h sup t ∈ [0 ,T ∧ e τ ǫN ] k e Z ǫt k H i + θ E Z T ∧ e τ ǫN k e Z ǫt k γ V dt ≤ C N,M,T n ǫ sup t ∈ [0 ,T ] (cid:0) k ¯ X φt k H (cid:1) + (1 + k x k H + k y k H ) h(cid:16) αǫ (cid:17) + δ i + E h sup t ∈ [0 ,T ∧ e τ ǫN ] (cid:12)(cid:12)(cid:12) Z t h G ( ¯ X φs )( φ ǫs − φ s ) , e Z ǫs i H ds (cid:12)(cid:12)(cid:12)i + √ ǫ E h sup t ∈ [0 ,T ∧ e τ ǫN ] (cid:12)(cid:12)(cid:12) Z t h e Z ǫt , G ( X ǫ,α,φ ǫ t ) dW t i H (cid:12)(cid:12)(cid:12)i + ǫ E Z T ∧ e τ ǫN k G ( X ǫ,α,φ ǫ t ) k L ( U,H ) dt + E h sup t ∈ [0 ,T ∧ e τ ǫN ] (cid:12)(cid:12)(cid:12) Z t h F ( X ǫ,α,φ ǫ s ( δ ) , b Y ǫ,αs ) − ¯ F ( X ǫ,α,φ ǫ s ( δ ) ) , e Z ǫs ( δ ) i H ds (cid:12)(cid:12)(cid:12)io . (4.12)Making use of Burkholder-Davis-Gundy’s inequality and Young’s inequality implies that C N,M,T n √ ǫ E h sup t ∈ [0 ,T ∧ e τ ǫN ] (cid:12)(cid:12)(cid:12) Z t h e Z ǫt , G ( X ǫ,α,φ ǫ t ) dW t i H (cid:12)(cid:12)(cid:12)i + ǫ E Z T ∧ e τ ǫN k G ( X ǫ,α,φ ǫ t ) k L ( U,H ) dt o ≤ C N,M,T √ ǫ E (cid:16) Z T ∧ e τ ǫN k e Z ǫt k H k G ( X ǫ,α,φ ǫ t ) k L ( U,H ) dt (cid:17) + C N,M,T ǫ E Z T ∧ e τ ǫN k G ( X ǫ,α,φ ǫ t ) k L ( U,H ) dt ≤ C N,M,T √ ǫ E (cid:16) sup t ∈ [0 ,T ∧ e τ ǫN ] k e Z ǫt k H × Z T ∧ e τ ǫN k G ( X ǫ,α,φ ǫ t ) k L ( U,H ) dt (cid:17) + C N,M,T ǫ E Z T ∧ e τ ǫN k G ( X ǫ,α,φ ǫ t ) k L ( U,H ) dt ≤ E h sup t ∈ [0 ,T ∧ e τ ǫN ] k e Z ǫt k H i + C N,M,T ǫ E Z T ∧ e τ ǫN (1 + k X ǫ,α,φ ǫ t k H ) dt ≤ E h sup t ∈ [0 ,T ∧ e τ ǫN ] k e Z ǫt k H i + C N,M,T ǫ. (4.13)23herefore, substituting (4.13) into (4.12) leads to E h sup t ∈ [0 ,T ∧ e τ ǫN ] k e Z ǫt k H i + 2 θ E Z T ∧ e τ ǫN k e Z ǫt k γ V dt ≤ C N,M,T n ǫ sup t ∈ [0 ,T ] (cid:0) k ¯ X φt k H (cid:1) + (1 + k x k H + k y k H ) h(cid:16) αǫ (cid:17) + δ i + E h sup t ∈ [0 ,T ∧ e τ ǫN ] (cid:12)(cid:12)(cid:12) Z t h G ( ¯ X φs )( φ ǫs − φ s ) , e Z ǫs i H ds (cid:12)(cid:12)(cid:12)i + E h sup t ∈ [0 ,T ∧ e τ ǫN ] (cid:12)(cid:12)(cid:12) Z t h F ( X ǫ,α,φ ǫ s ( δ ) , b Y ǫ,αs ) − ¯ F ( X ǫ,α,φ ǫ s ( δ ) ) , e Z ǫs ( δ ) i H ds (cid:12)(cid:12)(cid:12)io . (4.14) Step 2 : In this step, we aim to estimate the term E h sup t ∈ [0 ,T ∧ e τ ǫN ] (cid:12)(cid:12)(cid:12) R t h G ( ¯ X φs )( φ ǫs − φ s ) , e Z ǫs i H ds (cid:12)(cid:12)(cid:12)i in (4.14). Firstly, it is easy to see that E h sup t ∈ [0 ,T ∧ e τ ǫN ] (cid:12)(cid:12)(cid:12) Z t h G ( ¯ X φs )( φ ǫs − φ s ) , e Z ǫs i H ds (cid:12)(cid:12)(cid:12)i ≤ X i =1 e I i ( N, ǫ ) + E (cid:0) b I ( N, ǫ ) (cid:1) , (4.15)where we denote e I ( N, ǫ ) := E h sup t ∈ [0 ,T ∧ e τ ǫN ] (cid:12)(cid:12)(cid:12) Z t h G ( ¯ X φs )( φ ǫs − φ s ) , e Z ǫs − e Z ǫs ( δ ) i H ds (cid:12)(cid:12)(cid:12)i , e I ( N, ǫ ) := E h sup t ∈ [0 ,T ∧ e τ ǫN ] (cid:12)(cid:12)(cid:12) Z t h (cid:0) G ( ¯ X φs ) − G ( ¯ X φs ( δ ) ) (cid:1) ( φ ǫs − φ s ) , e Z ǫs ( δ ) i H ds (cid:12)(cid:12)(cid:12)i , e I ( N, ǫ ) := E h sup t ∈ [0 ,T ∧ e τ ǫN ] (cid:12)(cid:12)(cid:12) Z tt ( δ ) h G ( ¯ X φs ( δ ) )( φ ǫs − φ s ) , e Z ǫs ( δ ) i H ds (cid:12)(cid:12)(cid:12)i , b I ( N, ǫ ) := [( T ∧ e τ ǫN ) /δ ] − X k =0 (cid:12)(cid:12)(cid:12) h G ( ¯ X φkδ ) Z ( k +1) δkδ ( φ ǫs − φ s ) ds, e Z ǫkδ i H (cid:12)(cid:12)(cid:12) . From Cauchy-Schwarz’s inequality, condition ( A3 ), Lemma 3.5 and Lemma 4.3, it followsthat e I ( N, ǫ ) ≤ E h Z T ∧ e τ ǫN k G ( ¯ X φt ) k L ( U,H ) k φ ǫt − φ t k U k e Z ǫt − e Z ǫt ( δ ) k H dt i ≤ n E h Z T ∧ e τ ǫN k G ( ¯ X φt ) k L ( U,H ) k φ ǫt − φ t k U dt io n E h Z T ∧ e τ ǫN k e Z ǫt − e Z ǫt ( δ ) k H dt io ≤ n E h Z T ∧ e τ ǫN C (1 + k ¯ X φt k H ) k φ ǫt − φ t k U dt io × n E h Z T ∧ e τ ǫN k ¯ X φt − ¯ X φt ( δ ) k H + k X ǫ,α,φ ǫ t − X ǫ,α,φ ǫ t ( δ ) k H ) dt io ≤ C N δ (1 + k x k H + k y k H ) n E h Z T k φ ǫt − φ t k U dt io ≤ C N,M δ (1 + k x k H + k y k H ) . (4.16)24y Lemma 4.3, the second term can be controlled as follows e I ( N, ǫ ) ≤ E h Z T ∧ e τ ǫN k G ( ¯ X φt ) − G ( ¯ X φt ( δ ) ) k L ( U,H ) k φ ǫt − φ t k U k e Z ǫt ( δ ) k H dt i ≤ n E h Z T ∧ e τ ǫN k ¯ X φt − ¯ X φt ( δ ) k H k X ǫ,α,φ ǫ t ( δ ) − ¯ X φt ( δ ) k H dt io n E h Z T k φ ǫt − φ t k U dt io ≤ C N,M δ (1 + k x k H + k y k H ) . (4.17)Using H¨older’s inequality twice, we obtain e I ( N, ǫ ) ≤ n E sup t ∈ [0 ,T ∧ e τ ǫN ] (cid:12)(cid:12)(cid:12) Z tt ( δ ) k G ( ¯ X φs ( δ ) ) k L ( U,H ) k φ ǫs − φ s k U ds (cid:12)(cid:12)(cid:12) o × n E h sup t ∈ [0 ,T ∧ e τ ǫN ] k X ǫ,α,φ ǫ t − ¯ X φt k H io ≤ δ n E Z T ∧ e τ ǫN (1 + k ¯ X φs ( δ ) k H ) k φ ǫs − φ s k U ds o n E h sup t ∈ [0 ,T ∧ e τ ǫN ] k X ǫ,α,φ ǫ t − ¯ X φt k H io ≤ C N δ n E h Z T k φ ǫt − φ t k U dt io ≤ C N,M δ . (4.18)Now let us consider the convergence of the last term b I ( N, ǫ ). Since A M is a Polishspace and { φ ǫ : ǫ > } ⊂ A M converges to φ in distribution as S M -valued random elements,we are able to use the Skorokhod representation theorem to construct a probability space (cid:16)e Ω , f F , f F t ≥ , e P (cid:17) and processes ( e φ ǫ , e φ, f W ǫ ) such that the joint distribution of ( e φ ǫ , f W ǫ ) is thesame as ( φ ǫ , W ǫ ) and e φ ǫ → e φ , e P -a.s., in the weak topology of S M , where W ǫ is defined in(3.10). Therefore, for each a, b ∈ [0 , T ], a < b , the integral R ba e φ ǫs ds → R ba e φ s ds weakly in U .Without loss of generality, we will use the notations (Ω , F , F t ≥ , P ) and ( φ ǫ , φ, W ) replacing (cid:16)e Ω , f F , f F t ≥ , e P (cid:17) and ( e φ ǫ , e φ, f W ǫ ), respectively.Since G ( ¯ X φkδ ) is a Hilbert-Schmidt operator hence is compact operator, we infer that (cid:13)(cid:13)(cid:13) G ( ¯ X φkδ ) (cid:16) Z ( k +1) δkδ φ ǫs ds − Z ( k +1) δkδ φ s ds (cid:17)(cid:13)(cid:13)(cid:13) H → , as ǫ → , which implies that b I ( N, ǫ, ω ) → P -a.s. as ǫ →
0. Furthermore, it is easy to see thatfor any fixed
N > b I ( N, ǫ, ω ) ≤ C N,M by the similar arguments as in (4.17), then thedominated convergence theorem yields that for any
N > E (cid:0) b I ( N, ǫ ) (cid:1) → , as ǫ → . (4.19)Finally, taking (4.16)-(4.19) into (4.15) account we conclude that for any fixed N > δ → lim sup ǫ → E h sup t ∈ [0 ,T ∧ e τ ǫN ] (cid:12)(cid:12)(cid:12) Z t h G ( ¯ X φs )( φ ǫs − φ s ) , e Z ǫs i H ds (cid:12)(cid:12)(cid:12)i = 0 . (4.20)25 tep 3 : This step is devoted to investigating the term E h sup t ∈ [0 ,T ∧ e τ ǫN ] (cid:12)(cid:12)(cid:12) R t h F ( X ǫ,α,φ ǫ s ( δ ) , b Y ǫ,αs ) − ¯ F ( X ǫ,α,φ ǫ s ( δ ) ) , e Z ǫs ( δ ) i H ds (cid:12)(cid:12)(cid:12)io in (4.14), which is similar to the recent work [27], we include ithere for the completeness.It is obvious that (cid:12)(cid:12)(cid:12) Z t h F ( X ǫ,α,φ ǫ s ( δ ) , b Y ǫ,αs ) − ¯ F ( X ǫ,α,φ ǫ s ( δ ) ) , e Z ǫs ( δ ) i H ds (cid:12)(cid:12)(cid:12) ≤ [ t/δ ] − X k =0 (cid:12)(cid:12)(cid:12) Z ( k +1) δkδ h F ( X ǫ,α,φ ǫ s ( δ ) , b Y ǫ,αs ) − ¯ F ( X ǫ,α,φ ǫ s ( δ ) ) , e Z ǫs ( δ ) i H ds (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) Z tt ( δ ) h F ( X ǫ,α,φ ǫ s ( δ ) , b Y ǫ,αs ) − ¯ F ( X ǫ,α,φ ǫ s ( δ ) ) , e Z ǫs ( δ ) i H ds (cid:12)(cid:12)(cid:12) =: J ( t ) + J ( t ) . (4.21)According to the condition ( A2 ), the Lipschitz continuity of ¯ F and the definition of e τ ǫN , theterm J ( t ) can be controlled by E h sup t ∈ [0 ,T ∧ e τ ǫN ] J ( t ) i ≤ h E (cid:16) sup t ∈ [0 ,T ∧ e τ ǫN ] k X ǫ,α,φ ǫ t − ¯ X φt k H (cid:17)i × h E (cid:16) sup t ∈ [0 ,T ∧ e τ ǫN ] (cid:12)(cid:12)(cid:12) Z tt ( δ ) (1 + k X ǫ,α,φ ǫ s ( δ ) k H + k b Y ǫ,αs k H ) ds (cid:12)(cid:12)(cid:12) (cid:17)i ≤ n E (cid:16) sup t ∈ [0 ,T ∧ e τ ǫN ] k X ǫ,α,φ ǫ t − ¯ X φt k H (cid:17)o × n E h Z T ∧ e τ ǫN (1 + k X ǫ,α,φ ǫ t ( δ ) k H ) dt i + sup t ∈ [0 ,T ] E k b Y ǫ,αt k H o δ ≤ C N,T (1 + k x k H + k y k H ) δ , (4.22)where we used Lemma 4.1 in the last step.The term J ( t ) will be controlled as follows. E h sup t ∈ [0 ,T ∧ e τ ǫN ] J ( t ) i ≤ E [ T ∧ e τ ǫN /δ ] − X k =0 (cid:12)(cid:12)(cid:12) Z ( k +1) δkδ h F ( X ǫ,α,φ ǫ kδ , b Y ǫ,αs ) − ¯ F ( X ǫ,α,φ ǫ kδ ) , e Z ǫkδ i H ds (cid:12)(cid:12)(cid:12) ≤ C T δ sup ≤ k ≤ [ T ∧ e τ ǫN /δ ] − E (cid:12)(cid:12)(cid:12) Z ( k +1) δkδ h F ( X ǫ,α,φ ǫ kδ , b Y ǫ,αs ) − ¯ F ( X ǫ,α,φ ǫ kδ ) , e Z ǫkδ i H ds (cid:12)(cid:12)(cid:12) ≤ C T αδ sup ≤ k ≤ [ T ∧ e τ ǫN /δ ] − (cid:16) E k X ǫ,α,φ ǫ kδ − ¯ X φkδ k H (cid:17) (cid:16) E (cid:13)(cid:13)(cid:13) Z δα F ( X ǫ,α,φ ǫ kδ , b Y ǫ,αsα + kδ ) − ¯ F ( X ǫ,α,φ ǫ kδ ) ds (cid:13)(cid:13)(cid:13) H (cid:17) ≤ C N,T αδ sup ≤ k ≤ [ T ∧ e τ ǫN /δ ] − (cid:16) Z δα Z δα r Ψ k ( s, r ) dsdr (cid:17) , where for each 0 ≤ r ≤ s ≤ δα ,Ψ k ( s, r ) := E h h F ( X ǫ,α,φ ǫ kδ , b Y ǫ,αsα + kδ ) − ¯ F ( X ǫ,α,φ ǫ kδ ) , F ( X ǫ,α,φ ǫ kδ , b Y ǫ,αrα + kδ ) − ¯ F ( X ǫ,α,φ ǫ kδ ) i H i . k ( s, r ) ≤ C T (1 + k x k H + k y k H ) e − ( s − r ) ̺ , where ̺ > E h sup t ∈ [0 ,T ∧ e τ ǫN ] J ( t ) i ≤ C N,T (1 + k x k H + k y k H ) αδ (cid:16) Z δα Z δα r e − ( s − r ) ̺ dsdr (cid:17) ≤ C N,T (1 + k x k H + k y k H ) αδ (cid:16) δα̺ − ̺ + 1 ̺ e − ̺δ α (cid:17) . (4.23)Combining (4.22), (4.23) with (4.21) implies that E h sup t ∈ [0 ,T ∧ e τ ǫN ] (cid:12)(cid:12)(cid:12) Z t h F ( X ǫ,α,φ ǫ s ( δ ) , b Y ǫ,αs ) − ¯ F ( X ǫ,α,φ ǫ s ( δ ) ) , e Z ǫs ( δ ) i H ds (cid:12)(cid:12)(cid:12)i ≤ C N,T (1 + k x k H + k y k H ) (cid:16) αδ + α δ + δ (cid:17) . (4.24) Step 4 : After all of preparations above, we are in a position to derive the desired resultson the convergence of e Z ǫt in distribution.For any ε >
0, using Chebyshev’s inequality we obtain that P n(cid:16) sup t ∈ [0 ,T ] k e Z ǫt k H + θ Z T k e Z ǫt k γ V dt (cid:17) > ε o ≤ ε E h(cid:16) sup t ∈ [0 ,T ] k e Z ǫt k H + θ Z T k e Z ǫt k γ V dt (cid:17) { T ≤ e τ ǫN } i + 1 ε E h(cid:16) sup t ∈ [0 ,T ] k e Z ǫt k H + θ Z T k e Z ǫt k γ V dt (cid:17) { T > e τ ǫN } i . (4.25)We focus on the second term of right hand side of (4.25), using Markov’s inequality andH¨older’s inequality, 1 ε E h(cid:16) sup t ∈ [0 ,T ] k e Z ǫt k H + θ Z T k e Z ǫt k γ V dt (cid:17) { T > e τ ǫN } i ≤ ε h E (cid:16) sup t ∈ [0 ,T ] k e Z ǫt k H + θ Z T k e Z ǫt k γ V dt (cid:17)i h P ( T > e τ ǫN ) i ≤ C (1 + k x k H + k y k H ) ε √ N , (4.26)where we used Lemma 3.3 and Lemma 3.4 in the last step.Finally, letting δ := α and collecting (4.14), (4.20) and (4.24), by the condition (2.8),we can get thatlim sup ǫ → P n sup t ∈ [0 ,T ] k e Z ǫt k H + θ Z T k e Z ǫt k γ V dt > ε o ≤ C (1 + k x k H + k y k H ) √ N , where the constant C does not depend on N , which implies the desired assertion by taking N → ∞ . We complete the proof of Condition (A) (i). (cid:3) .3 Compactness This subsection is devoted to concerning the compactness result, which also shows therate function I defined in (2.4) is a good rate function. After that, combining Theorem 4.1,we conclude that the random variable family { X ǫ,α } satisfies the Laplace principle (Theorem2.1), which is equivalent to the LDP on C ([0 , T ]; H ) ∩ L γ ([0 , T ]; V ). Theorem 4.2
Assume that the conditions given in Theorem 2.1 hold. For fixed
M > , x ∈ H and y ∈ H , let K M = { ¯ X φ : φ ∈ S M } , here ¯ X φ is a unique solution to the skeletonequation (2.7). Then K M is a compact set of C ([0 , T ]; H ) ∩ L γ ([0 , T ]; V ) .Proof Let us define any sequence { ¯ X φ n } in K M , which is the solution of Eq. (2.7) with φ n ∈ S M instead of φ , i.e., d ¯ X φ n t dt = (cid:2) A ( ¯ X φ n t ) + ¯ F ( ¯ X φ n t ) (cid:3) + G ( ¯ X φ n t ) φ nt , ¯ X φ n = x ∈ H . Noting that S M is a bounded closed subset in L ([0 , T ]; U ), hence is weakly compact, thereexists a subsequence also denoted by φ n , which weakly converges to a limit φ ∈ S M in L ([0 , T ]; U ). Then the prior estimates formulated in Lemma 3.3 imply that¯ X φ n → ¯ X φ weakly star in L ∞ ([0 , T ]; H ) , ¯ X φ n → ¯ X φ weakly in L γ ([0 , T ]; V ) . It is easy to show that ¯ X φ is the unique solution of the following limit equation d ¯ X φt dt = (cid:2) A ( ¯ X φt ) + ¯ F ( ¯ X φt ) (cid:3) + G ( ¯ X φt ) φ t , ¯ X φ = x. In order to study the compactness of set K M , it suffices to prove that ¯ X φ n strong convergesto ¯ X φ in C ([0 , T ]; H ) ∩ L γ ([0 , T ]; V ) as n → ∞ . Denote b Z nt := ¯ X φ n t − ¯ X φt fulfilling d b Z nt dt = (cid:2) A ( ¯ X φ n t ) − A ( ¯ X φt ) + ¯ F ( ¯ X φ n t ) − ¯ F ( ¯ X φt ) (cid:3) dt + (cid:2) G ( ¯ X φ n t ) φ nt − G ( ¯ X φt ) φ t (cid:3) dt, b Z n = 0 . It is easy to get the following energy estimate by the condition ( A2 ) and Young’s inequality, k b Z nt k H + θ Z t k b Z ns k γ V ds ≤ Z t (cid:0) C + ρ ( ¯ X φs ) + k φ ns k U (cid:1) k b Z ns k H ds +2 Z t h G ( ¯ X φs )( φ ns − φ s ) , b Z ns i H ds, where C > n . The prior estimate (3.7) implies that thereexist some constant K > n such thatsup n h sup t ∈ [0 ,T ] ( k ¯ X φ n t k H + k ¯ X φt k H ) + θ Z T ( k ¯ X φ n t k γ V + k ¯ X φt k γ V ) dt i = K . (4.27)Therefore, making use of Gronwall’s lemma and (4.27), it follows thatsup t ∈ [0 ,T ] k b Z nt k H + θ Z T k b Z nt k γ V dt ≤ C K ,M,T X i =1 I n,i , (4.28)28ere we denote I n, := Z T (cid:12)(cid:12)(cid:12) h G ( ¯ X φs )( φ ns − φ s ) , b Z ns − b Z ns ( δ ) i H (cid:12)(cid:12)(cid:12) ds,I n, := Z T (cid:12)(cid:12)(cid:12) h (cid:0) G ( ¯ X φs ) − G ( ¯ X φs ( δ ) ) (cid:1) ( φ ns − φ s ) , b Z ns ( δ ) i H (cid:12)(cid:12)(cid:12) ds,I n, := sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12) Z tt ( δ ) h G ( ¯ X φs ( δ ) )( φ ns − φ s ) , b Z ns ( δ ) i H ds (cid:12)(cid:12)(cid:12) ,I n, := sup t ∈ [0 ,T ] [ t/δ ] − X k =0 (cid:12)(cid:12)(cid:12) h G ( ¯ X φkδ ) Z ( k +1) δkδ ( φ ns − φ s ) ds, b Z nkδ i H (cid:12)(cid:12)(cid:12) . Following the almost same arguments as in the proof of Theorem 4.1, one can obtain I n, ≤ n Z T C (1 + k ¯ X φt k H ) k φ nt − φ t k U dt o n Z T k ¯ X φt − ¯ X φt ( δ ) k H + k ¯ X φ n t − ¯ X φ n t ( δ ) k H ) dt o ≤ C K ,M δ (1 + k x k H ) , (4.29) I n, ≤ n Z T k φ nt − φ t k U dt o n Z T k ¯ X φt − ¯ X φt ( δ ) k H k ¯ X φ n t ( δ ) − ¯ X φt ( δ ) k H dt o ≤ C K ,M δ (1 + k x k H ) , (4.30) I n, ≤ δ n Z T (1 + k ¯ X φt ( δ ) k H ) k φ nt − φ t k U dt o n sup t ∈ [0 ,T ] k ¯ X φ n t − ¯ X φt k H o ≤ C K ,M δ . (4.31)For the term I n, , since G ( ¯ X φkδ ) is a compact operator, the sequence G ( ¯ X φkδ ) R ( k +1) δkδ ( φ ns − φ s ) ds strongly converges to 0 in H for any fixed k , as n → ∞ . This combines with theboundedness of I n, implies that lim n →∞ I n, = 0. Furthermore, according to (4.29)-(4.31),for any δ >
0, lim n →∞ n sup t ∈ [0 ,T ] k b Z nt k H + θ Z T k b Z nt k γ V dt o ≤ Cδ , where the constant C > δ . Taking δ →
0, one can show that everysequence in K M has a convergent subsequence, therefore K M is a pre-compact subset of C ([0 , T ]; H ) ∩ L γ ([0 , T ]; V ). It suffices to prove that K M is a closed subset of C ([0 , T ]; H ) ∩ L γ ([0 , T ]; V ). It should be noted that the above arguments also implies that there existsa subsequence { ¯ X φ nk , k ≥ } converges to an element ¯ X φ ∈ K M in the same topology of C ([0 , T ]; H ) ∩ L γ ([0 , T ]; V ), which yields the desired results. We complete the proof of Condition (A) (ii). (cid:3)
We are now in a position to complete the proof of our main results in this paper.
Proof of Theorem 2.1 : Following from Theorem 4.1 and Theorem 4.2, we can infer thatthe random variable family { X ǫ,α } fulfills the Laplace principle, which is equivalent to theLDP on C ([0 , T ]; H ) ∩ L γ ([0 , T ]; V ) with a good rate function I defined in (2.4). (cid:3) Our general results formulated in Theorem 2.1 and 2.2 can be used to deal with a verylarge family of SPDE models directly, which not only extends or improves some of existing29orks using mild solution approach for a class of two-time-scale semi-linear SPDEs such asstochastic reaction-diffusion equations, stochastic Burgers equations (see e.g. [35]), but alsoobtain the LDP for several new infinite-dimensional models with respect to the two-time-scale case.In this section, we will denote by Λ ⊆ R d an open bounded domain with a smoothboundary. Let C ∞ (Λ , R d ) be the space of all infinitely differentiable functions from Λ to R d with compact support. For p ≥
1, let L p (Λ , R d ) denote the vector valued L p -space with thenorm k · k L p . For each integer m >
0, we use W m,p (Λ , R d ) to denote the classical Sobolevspace defined on Λ taking values in R d with respect to the (equivalent) norm: k u k W m,p = X ≤| α |≤ m Z Λ | D α u | p dx p . Below we would like to recall the so-called Gagliardo-Nirenberg interpolation inequality(cf. [36, Theorem 2.1.5]) for the reader’s convenience.If for any m, n ∈ N and q ∈ [1 , ∞ ] satisfying1 q = 12 + nd − mθd , nm ≤ θ ≤ , then there is a constant C > k u k W n,q ≤ C k u k θW m, k u k − θL , u ∈ W m, (Λ , R d ) . (5.1) Let us denote by ( E, M , m ) a separable probability space and ( L, D ( L )) a negativedefinite linear self-adjoint map defined on ( L ( m ) , h· , ·i ), which has discrete spectrum witheigenvalues 0 > − λ ≥ − λ ≥ · · · → −∞ . Let H be the topological dual space of D ( √− L ), which is endowed with the scalar product h u, v i H := Z E (cid:0) √− Lu ( ξ ) (cid:1) · (cid:0) √− Lv ( ξ ) (cid:1) dξ, u, v ∈ H , then identify L ( m ) with its dual, one can obtain the following dense and continuous em-bedding D ( √− L ) ⊆ L ( m ) ⊆ H . Consequently, due to this embedding, we can define V := D ( √− L ) , H := L ( m ) . Suppose that L − is continuous in V := L r +1 ( m ), where r > V ∗ by the following embedding V ⊂ H ∼ = D ( √− L ) ⊂ V ∗ , where ∼ = is understood through √− L . 30onsider the two-time-scale stochastic porous media equation as follows: dX ǫ,αt = [ L Ψ( X ǫ,αt ) + Φ( X ǫ,αt )] dt + F ( X ǫ,αt , Y ǫ,αt ) dt + √ ǫG ( X ǫ,αt ) dW t ,dY ǫ,αt = 1 α [ LY ǫ,αt + F ( X ǫ,αt , Y ǫ,αt )] dt + 1 √ α G ( X ǫ,αt , Y ǫ,αt ) dW t ,X ǫ,α = x ∈ H , Y ǫ,α = y ∈ H , (5.2)here W t is a cylindrical Wiener process defined on a probability space (Ω , F , F t , P ) takingvalues in a sparable Hilbert space U , Ψ , Φ : R → R are continuous and measurable mapssuch that there exist some constants θ > K , | Ψ( s ) | + | Φ( s ) | ≤ K (1 + | s | r ) , s ∈ R ; (5.3) −h Ψ( u ) − Ψ( v ) , u − v i − h Φ( u ) − Φ( v ) , L − ( u − v ) i≤ − θ k u − v k r +1 V + K k u − v k H , u, v ∈ V , (5.4)and the measurable maps F : H × H → H , G : V → L ( U, H ) , F : H × V → V ∗ , G : H × V → L ( U, H )are Lipschitz continuous, i.e., k F ( u , v ) − F ( u , v ) k H ≤ C (cid:0) k u − u k H + k v − v k H (cid:1) , (5.5) k G ( u ) − G ( v ) k L ( U,H ) ≤ C k u − v k H , (5.6) k F ( u , v ) − F ( u , v ) k H ≤ C k u − u k H + L F k v − v k H , (5.7) k G ( u , v ) − G ( u , v ) k L ( U,H ) ≤ C k u − u k H + L G k v − v k H , (5.8)here L F and L G represent the Lipschitz constants with respect to second variable of F and G , respectively. Furthermore, we assume thatsup v ∈ V k G ( u , v ) k L ( U,H ) ≤ C (1 + k u k H ) , (5.9)and the smallest eigenvalue λ of map L satisfies λ − L F − L G > . (5.10) Theorem 5.1 (stochastic porous media equation) Assume that Ψ , Φ satisfy the above con-ditions (5.3)-(5.4) and F , F , G , G satisfy (5.5)-(5.10), then the family { X ǫ,α : ǫ > } in(5.2) satisfies the LDP on C ([0 , T ]; H ) ∩ L r +1 ([0 , T ]; V ) with the good rate function I givenby (2 . .Proof It is well-established that the map A := L Ψ + Φ satisfy ( A1 )-( A3 ) for ρ ≡ , β =0 , γ = r +1, we refer to [24, Example 4.1.11] for some details. Moreover, one can easily provethat the assumptions presented in Theorem 2.1 hold via (5.5)-(5.10). Therefore, Theorem5.1 is a direct consequence of Theorem 2.1. (cid:3) Remark 5.1 (i) A typical example is that L = ∆ , the Laplace operator on a smooth boundeddomain in a complete Riemannian manifold with Dirichlet boundary, and Ψ( s ) := | s | r − s, Φ( s ) := s, s ∈ R . (ii) To the best of our knowledge, there is no result concerning the LDP for multi-time-scale stochastic porous media equation, even quasi-linear SPDEs (see Section 5.2 and 5.3 forother types of quasi-linear SPDEs), thus Theorem 5.1 and Theorem 5.2, 5.3 below seem tobe new in the literature. .2 Stochastic p -Laplace equation We consider the following Gelfand triple for the slow component V := W ,p (Λ) ⊂ H := L (Λ) ⊂ ( W ,p (Λ)) ∗ = V ∗ and the Gelfand triple for the fast component V := W , (Λ) ⊂ H := L (Λ) ⊂ ( W , (Λ)) ∗ = V ∗ . We introduce the two-time-scale stochastic p -Laplace equation as follows dX ǫ,αt = [ div ( |∇ X ǫ,αt | p − ∇ X ǫ,αt ) − C | X ǫ,αt | q − X ǫ,αt ] dt + F ( X ǫ,αt , Y ǫ,αt ) dt + √ ǫG ( X ǫ,αt ) dW t ,dY ǫ,αt = 1 α [∆ Y ǫ,αt + F ( X ǫ,αt , Y ǫ,αt )] dt + 1 √ α G ( X ǫ,αt , Y ǫ,αt ) dW t ,X ǫ,α = x ∈ H , Y ǫ,α = y ∈ H , (5.11)where C >
0, 2 ≤ p ≤ ∞ , ≤ q ≤ p and W t is a cylindrical Wiener process in U defined ona probability space (Ω , F , F t , P ). Theorem 5.2 (stochastic p -Laplace equation) Assume that F , F , G , G satisfy (5.5)-(5.10)with ∆ replacing operator L , then the family { X ǫ,α : ǫ > } in (5.11) satisfies the LDP on C ([0 , T ]; H ) ∩ L p ([0 , T ]; V ) with the good rate function I given by (2 . .Proof It is well-known that the p -Laplace operator satisfies the hemicontinuity, classicalmonotonicity and growth condition ( A1 )-( A3 ) for ρ ≡ , β = 0 , γ = p , we can see e.g. [23,Example 5.5] for the precise proof. By (5.5)-(5.10), one can easily check that the assumptionspresented in Theorem 2.1 hold. Thus, Theorem 5.2 follows from the result obtain in Theorem2.1. (cid:3) Remark 5.2
This theorem can not be applied to the case of < p < directly, however, weare able to use Theorem 2.2 to derive the LDP for (5.11) on C ([0 , T ]; H ) (cf. [24, Example4.1.9]). Suppose the same setting as in Section 5.1 for the case of 0 < r <
1, the two-time-scalestochastic fast-diffusion equation is given by dX ǫ,αt = [ L Ψ( X ǫ,αt ) + F ( X ǫ,αt , Y ǫ,αt )] dt + √ ǫG ( X ǫ,αt ) dW t ,dY ǫ,αt = 1 α [ LY ǫ,αt + F ( X ǫ,αt , Y ǫ,αt )] dt + 1 √ α G ( X ǫ,αt , Y ǫ,αt ) dW t ,X ǫ,α = x ∈ H , Y ǫ,α = y ∈ H , (5.12)here W t stands for a cylindrical Wiener process defined on a probability space (Ω , F , F t , P )taking values in a sparable Hilbert space U , Ψ : R → R is continuous and measurable mapsuch that there exist some constants δ > K , | Ψ( s ) | ≤ K (1 + | s | r ) , s ∈ R ; (5.13)(Ψ( s ) − Ψ( s ))( s − s ) ≥ δ | s − s | ( | s | ∨ | s | ) r − , s , s ∈ R . (5.14)32 heorem 5.3 (stochastic fast-diffusion equation) Suppose that Ψ satisfies the conditions(5.13)-(5.14) and F , F , G , G satisfy (5.5)-(5.10) above, then the family { X ǫ,α : ǫ > } in(5.12) satisfies the LDP on C ([0 , T ]; H ) with the good rate function I given by (2 . .Proof Following the similar arguments as in Section 5.1, the map A := L Ψ satisfies con-ditions ( A1 ), ( A3 ) and ( A4 ) for ρ ≡ , β = 0 , γ = r + 1, one can see also [24, Example4.1.11] for the detailed calculations. According to (5.5)-(5.10), the assumptions given inTheorem 2.1 hold. Finally, the assertion formulated in Theorem 5.3 follows from Theorem2.2. (cid:3) Remark 5.3 (i) A specific example fulfilling (5.13)-(5.14) is that Ψ( s ) := | s | r − s , s ∈ R for < r < , which characterizes the classical fast-diffusion equation.(ii) In this case, for simplicity, we consider the situation that the embedding L r +1 ( m ) ⊂ H is continuous and dense, one can see [24, Remark 4.1.15] for the sufficient condition toguarantee such assumption holds. Besides the above two-time-scale quasi-linear type SPDEs, our main results of this paperare also applicable to a large class of semi-linear SPDEs only satisfying local monotonicityand polynomial growth conditions, for instance, the stochastic Burgers equation, stochasticNavier-Stokes equation and other hydrodynamical type models, which has been studied in[27] on the LDP for two-time-scale stochastic Burgers equation using mild solution method.In the following, we also extend the work of [27] on two-time-scale stochastic Burgers equationto more general case, and it should be mentioned that in this paper, instead of the mildsolution approach, we apply the variational approach to get the LDP, which helps us tocover more concrete examples.
The first semi-linear example is the two-time-scale stochastic multidimensional Burgerstype equation. Consider the Gelfand triple for the slow component V := W , (Λ) ⊂ H := L (Λ) ⊂ ( W , (Λ)) ∗ = V ∗ and also the Gelfand triple for the fast component V := W , (Λ) ⊂ H := L (Λ) ⊂ ( W , (Λ)) ∗ = V ∗ . The following two-time-scale stochastic Burgers type equation dX ǫ,αt = [∆ X ǫ,αt + h f ( X ǫ,αt ) , ∇ X ǫ,αt i + h ( X ǫ,αt )] dt + F ( X ǫ,αt , Y ǫ,αt ) dt + √ ǫG ( X ǫ,αt ) dW t ,dY ǫ,αt = 1 α [∆ Y ǫ,αt + F ( X ǫ,αt , Y ǫ,αt )] dt + 1 √ α G ( X ǫ,αt , Y ǫ,αt ) dW t ,X ǫ,α = x ∈ H , Y ǫ,α = y ∈ H , (5.15)where f = ( f , · · · , f d ) : R → R d is a Lipschitz continuous function and h· , ·i denotes thescalar product in R d , W t stands for a cylindrical Wiener process defined on a probabilityspace (Ω , F , F t , P ) taking values in a sparable Hilbert space U . Let h : R → R denote acontinuous function with h (0) = 0 such that for some constants C, r, s ∈ [0 , ∞ ) | h ( x ) | ≤ C ( | x | r + 1) , x ∈ R ; (5.16)( h ( x ) − h ( y ))( x − y ) ≤ C (1 + | y | s )( x − y ) , x, y ∈ R . (5.17)Now we state the main result on the LDP for two-time-scale stochastic multidimensionalBurgers type equation. 33 heorem 5.4 (stochastic Burgers type equation) Assume that h satisfies (5.16)-(5.17), F , F , G , G satisfy (5.5)-(5.10) above with ∆ replacing operator L . AssumeCase 1: d = 1 , r = 2 , s = 2 ,Case 2: d = 2 , r = 2 , s = 2 , and f is bounded,Case 3: d = 3 , r = 2 , s = , and f is bounded measurable function independent of X ǫ,αt .Then the family { X ǫ,α : ǫ > } in (5.15) satisfies the LDP on C ([0 , T ]; H ) ∩ L ([0 , T ]; V ) with the good rate function I given by (2 . .Proof Let us denote the operator A ( u ) := b A ( u ) + h ( u ) := ∆ u + h f ( u ) , ∇ u i + h ( u ) , u ∈ V . Following from [25, Example 3.2], it is easy to obtain that map A satisfies ( A1 )-( A2 ) with γ = 2. For the condition ( A3 ), we have k b A ( u ) + h ( u ) k V ∗ ≤ C ( k b A ( u ) k V ∗ + k h ( u ) k V ∗ ) . The first term of right hand side of the above inequality fulfills k b A ( u ) k V ∗ ≤ C (1 + k u k V ∗ )(1 + k u k νH ∗ ) , where ν = 2 in Case 1 and ν = 0 in Case 2. For the second term, making use of H¨older’sinequality and Gagliardo-Nirenberg interpolation inequality (5.1), we can get that | V ∗ h h ( u ) , v i V | ≤ k v k L ∞ (1 + k u k L ) , d = 1 , k v k L (1 + k u k L ) , d = 2 , k v k L (1 + k u k L ) , d = 3 ≤ k v k V (1 + k u k V k u k H ) . Then the condition ( A3 ) are satisfied with γ = β = 2. Consequently, the assertion followsby Theorem 2.1. (cid:3) Remark 5.4
If we take d = 1 , f ( x ) = x and h = 0 , Theorem 5.4 can be used to deal withthe classical stochastic Burgers equation. Moreover, it should be noted that one can alsoallow a polynomial control term h in the drift of (5.15). For instance, we may consider g ( x ) = − x + c x + c x ( c , c ∈ R ) and show that (5.16)-(5.17) are satisfied. Due to this,(5.15) also covers some of two-time-scale stochastic reaction-diffusion type equations. Below we are able to apply the main result in this work to several stochastic 2D hydro-dynamical systems.
The main idea of this subsection is to consider the two-time-scale stochastic 2D hydrody-namical type systems, which cover a wide class of mathematical coupled models from fluiddynamics.For the slow component, let H be a separable Hilbert space equipped with norm | · | , A be an (unbounded) positive linear self-adjoint operator on H . Define V = D ( A ), and theassociated norm k v k = | A v | for any v ∈ V . Let V ∗ be the dual space of V with respect tothe scalar product ( · , · ) on H . Due to this, one can consider a Gelfand triple V ⊂ H ⊂ V ∗ .Let us denote by h u, v i the dualization between u ∈ V and v ∈ V ∗ , and it is easy to seethat h u, v i = ( u, v ) if u ∈ V , v ∈ H . There exists an orthonormal basis { e k } k ≥ on H ofeigenfunctions of A , and the increasing eigenvalue sequence 0 < λ ≤ λ ≤ ... ≤ λ n ≤ ... ↑ ∞ . Let B : V × V → V ∗ be a continuous map fulfilling34C1) B : V × V → V ∗ is a continuous bilinear map.(C2) For all u i ∈ V , i = 1 , , h B ( u , u ) , u i = −h B ( u , u ) , u i , h B ( u , u ) , u i = 0 . (C3) There exists a Banach space H such that(i) V ⊂ H ⊂ H ;(ii) there exists a constant a > k u k H ≤ a | u |k u k for all u ∈ V ;(iii) for every η > C η > |h B ( u , u ) , u i| ≤ η k u k + C η k u k H k u k H for all u i ∈ V , i = 1 , , . For simplicity of notations, we denote B ( u ) := B ( u, u ). Moreover, we consider H := H , V := V and the Gelfand triple for the fast component V ⊂ H ⊂ V ∗ . The following is the two-time-scale stochastic 2D hydrodynamical type systems: dX ǫ,αt + [ AX ǫ,αt + B ( X ǫ,αt , X ǫ,αt )] dt = F ( X ǫ,αt , Y ǫ,αt ) dt + √ ǫG ( X ǫ,αt ) dW t ,dY ǫ,αt = 1 α [ AY ǫ,αt + F ( X ǫ,αt , Y ǫ,αt )] dt + 1 √ α G ( X ǫ,αt , Y ǫ,αt ) dW t ,X ǫ,α = x ∈ H , Y ǫ,α = y ∈ H . (5.18) Theorem 5.5 (stochastic 2D hydrodynamical type systems) Assume that B satisfies (C1)-(C3) and F , F , G , G satisfy (5.5)-(5.10) above with A replacing operator L , then thefamily { X ǫ,α : ǫ > } in (5.18) satisfies the LDP on C ([0 , T ]; H ) ∩ L ([0 , T ]; V ) with thegood rate function I given by (2 . .Proof It is suffices to check the conditions ( A1 )-( A3 ) hold for e A ( u ) := − Au − B ( u, u ).( A1 ): The hemicontinuity follows from the linearity and bilinearity of maps A and B ,respectively.( A2 ): It is easy to see that for all u, v ∈ V , h− Au − ( − Av ) , u − v i ≤ −k u − v k . (5.19)According to [6, Remark 2.1], we know that for any constant η > C η > u, v ∈ V |h B ( u ) − B ( v ) , u − v i| ≤ η k u − v k + C η | u − v | k v k H . (5.20)Thus the condition ( A2 ) follows by (5.19) and (5.20) with γ = 2.Furthermore, (2.7) in [6] implies that ( A3 ) holds with β = 2.Using (5.5)-(5.10), the rest of assumptions given in Theorem 2.1 are satisfied. Then theresult formulated in Theorem 5.5 is a consequent result of Theorem 2.1. (cid:3) emark 5.5 (i) The well-posedness and Freidlin-Wentzell type LDP of stochastic 2D hy-drodynamical type systems have been investigated by Chueshov and Millet in [6] using weakconvergence method in order to cover the important class of hydrodynamical models in bothbounded and unbounded domains. In this work, we are in a position to generalize the mainresults of [6] to the multi-scale case.(ii) Based on the work of [6], the main result obtained in this subsection is applicable tomany concrete hydrodynamical type systems, for instance, the stochastic 2D Navier-Stokesequation, stochastic 2D magneto-hydrodynamic equations, stochastic 2D Boussinesq equa-tions, stochastic 2D magnetic B´enard problem, stochastic 3D Leray- α model and also shellmodels of turbulence. We also refer the reader to [9, 15, 16] and references within for themathematical treatment and further studies of these models. The stochastic power law fluid equation characterizes the velocity field of a viscous andincompressible non-Newtonian fluids, which is an important model in hydrodynamical, onecan see [10, 28] and reference therein for more mathematical background of this type ofmodel.Let u : Λ → R d denote a vector field. Set e ( u ) : Λ → R d ⊗ R d ; e i,j ( u ) = ∂ i u j + ∂ j u i , i, j = 1 , · · · , d.τ ( u ) : Λ → R d ⊗ R d ; τ ( u ) = 2 ν (1 + | e ( u ) | ) p − e ( u ) , here ν > p > ∂ t u = div ( τ ( u )) − ( u · ∇ ) u − ∇ p + f, for div ( u ) = 0 , where u = u ( t, x ) = ( u i ( t, x )) di =1 and p stand for the velocity field and pressure of the fluid,respectively, f denotes the external force acting on the system, div ( τ ( u )) = d X j =1 ∂ j τ i,j ( u ) ! di =1 . Moreover, it should mentioned that if one take p = 2, the power law fluid equation reducesto the classical Navier-Stokes equation.For the slow component of two-time-scale situation, we consider the following Gelfandtriple: V ⊆ H ⊆ V ∗ , where we set V = n u ∈ W ,p (Λ; R d ) : div ( u ) = 0 o ; H = n u ∈ L (Λ; R d ) : div ( u ) = 0 o . Let P H denote a projection map onto H on L (Λ; R d ). Thus one can extend the operator A : W ,p (Λ; R d ) ∩ V → H , A ( u ) = P H [ div ( τ ( u ))]; F : (cid:0) W ,p (Λ; R d ) ∩ V (cid:1) × (cid:0) W ,p (Λ; R d ) ∩ V (cid:1) → H ; F ( u, v ) = − P H [( u · ∇ ) v ] , F ( u ) := F ( u, u )36o the map (see [26] for details): A : V → V ∗ ; F : V × V → V ∗ . Moreover, it is easy to show that hA ( u ) , v i V = − Z Λ d X i,j =1 τ i,j ( u ) e i,j ( v ) dx, u, v ∈ V ; V ∗ h F ( u, v ) , w i V = − V ∗ h F ( u, w ) , v i V , V ∗ h F ( u, v ) , v i V = 0 , u, v, w ∈ V . For the fast component of two-time-scale situation, we consider V := W , (Λ; R d ) ⊂ H := L (Λ; R d ) ⊂ ( W , (Λ; R d )) ∗ = V ∗ . Consequently, the multi-scale stochastic power law fluid equation can be written as followsin variational form: dX ǫ,αt = ( ν A X ǫ,αt + F ( X ǫ,αt ) + f ) dt + F ( X ǫ,αt , Y ǫ,αt ) dt + √ ǫG ( X ǫ,αt ) dW t ,dY ǫ,αt = 1 α [∆ Y ǫ,αt + F ( X ǫ,αt , Y ǫ,αt )] dt + 1 √ α G ( X ǫ,αt , Y ǫ,αt ) dW t ,X ǫ,α = x ∈ H , Y ǫ,α = y ∈ H . (5.21)for f ∈ H , W t is a cylindrical Wiener process on U . Theorem 5.6 (stochastic power law fluid equation) Let p ≥ d +22 and F , F , G , G satisfy(5.5)-(5.10) above with ∆ replacing operator L , then the family { X ǫ,α : ǫ > } in (5.21)satisfies the LDP on C ([0 , T ]; H ) ∩ L ([0 , T ]; V ) with the good rate function I given by (2 . ..Proof First, we want to check the conditions ( A1 )-( A3 ). Without loss of generality, wealways suppose that viscosity coefficient ν = 1. According to [28, Lemma 1.19], one showsthat Z Λ | e ( u ) | p dx ≥ C p k u k W ,p , u ∈ W ,p (Λ; R d ); d X i,j =1 τ i,j ( u ) e i,j ( u ) ≥ C ( | e ( u ) | p − d X i,j =1 ( τ i,j ( u ) − τ i,j ( v ))( e i,j ( u ) − e i,j ( v )) ≥ C ( | e ( u ) − e ( v ) | + | e ( u ) − e ( v ) | p ); | τ i,j ( u ) | ≤ C (1 + | e ( u ) | ) p − , i, j = 1 ..., d. Therefore, by means of the estimates above, for all u, v ∈ V , V ∗ h F ( u ) − F ( v ) , u − v i V = − V ∗ h F ( u − v ) , v i V = V ∗ h F ( u − v, v ) , u − v i V ≤ C k v k V k u − v k L pp − ≤ C k v k V k u − v k dp W , k u − v k p − dp H ≤ ε k u − v k W , + C ε k v k p p − d V k u − v k H .
37t follows that V ∗ hA ( u ) + F ( u ) − A ( v ) − F ( v ) , u − v i V = − Z Λ d X i,j =1 ( τ i,j ( u ) − τ i,j ( v ))( e i,j ( u ) − e i,j ( v )) dx ≤ − C k e ( u ) − e ( v ) k H ≤ − C k u − v k W , . Consequently, one can get that V ∗ hA ( u ) + F ( u ) − A ( v ) − F ( v ) i V ≤ − ( C − ε ) k u − v k W , + C ε k v k p p − d V k u − v k H , then the condition ( A2 ) is satisfied with ρ ( v ) = C ε k v k q q − d V and γ = p .For the condition ( A3 ), we infer that | V ∗ h F ( v ) , u i V | = | V ∗ h F ( v, u ) , v i V | ≤ k u k V k v k L pp − , u, v ∈ V , Then one can obtain k F ( v ) k V ∗ ≤ k v k L pp − , v ∈ V . Let q = dpd − p , γ = d ( d +2) p − d , using the Gagliardo-Nirenberg interpolation inequality (5.1) leadsto k v k L pp − ≤ k v k γL q k v k − γL ≤ C k v k γV k v k − γH . Since p ≥ d +22 , obviously, the condition ( A3 ) holds.By means of (5.5)-(5.10), the remainder of assumptions given in Theorem 2.1 hold. Con-sequently, Theorem 5.6 is a consequent result of Theorem 2.1. (cid:3) Remark 5.6 (i) The main results of this work on small noise type LDP for the multi-scalestochastic power law fluid equation seems not have been investigated in the literature.(ii) Besides the above examples, our general results are also applicable to the multi-scalestochastic Ladyzhenskaya model, which is a higher order variant of the power law fluid andpioneered by Ladyzhenskaya [21], see also [13, 43] for the further studies of this model. Weomit the detailed proof to keep down the size of this paper.
Acknowledgements : The research of S. Li is supported by NSFC (No. 12001247),NSF of Jiangsu Province (No. BK20201019), NSF of Jiangsu Higher Education Institu-tions of China (No. 20KJB110015) and the Foundation of Jiangsu Normal University(No. 19XSRX023). The research of W. Liu is supported by NSFC (No. 11822106, 11831014,12090011) and the PAPD of Jiangsu Higher Education Institutions.
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