Large deviations for locally monotone stochastic partial differential equations driven by Levy noise
aa r X i v : . [ m a t h . P R ] M a y Large deviations for locally monotone stochasticpartial differential equations driven by L´evy noise ∗ Jie Xiong † and Jianliang Zhai ‡ October 3, 2018
Abstract
In this paper, we establish a large deviation principle for a type of stochasticpartial differential equations (SPDEs) with locally monotone coefficients drivenby L´evy noise. The weak convergence method plays an important role.Keywords: Large Deviations, L´evy Processes, Monotone coefficients.2010 AMS Classification: Primary 60H15. Secondary 35R60, 37L55.
We shall prove via the weak convergence approach [7, 10, 17] the Freidlin-Wentzelltype large deviation principle (LDP) for a family of locally monotone stochastic par-tial differentia equations (SPDEs) driven by L´evy processes, these SPDEs includestochastic reaction-diffusion equations, stochastic Burgers type equations, stochastic2D Navier-Stokes equations and stochastic equations of non-Newtonian fluids.Let V be a reflexive and separable Banach space, which is densely and continuouslyinjected in a separable Hilbert space ( H, h· , ·i H ). Identifying H with its dual we get V ⊂ H ∼ = H ∗ ⊂ V ∗ , ∗ This work was started when the second author visited the first author at the Department ofMathematics, University of Macau. JX’s research is supported by Macao Science and Technol-ogy Fund FDCT 076/2012/A3 and Multi-Year Research Grants of the University of Macau Nos.MYRG2014-00015-FST and MYRG2014-00034-FST. JZ’s research is supported by National NaturalScience Foundation of China (NSFC) (No. 11431014, No. 11401557), and the Fundamental ResearchFunds for the Central Universities (No. WK 0010000048). † Department of Mathematics, Faculty of Sci. & Tech., University of Macau, Macau. Email: [email protected] ‡ School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026,China Email: [email protected] h· , ·i V ∗ ,V the duality between V ∗ and V , then we have h u, v i V ∗ ,V = h u, v i H , ∀ u ∈ H, v ∈ V. Fix
T > , F , ( F t ) t ∈ [0 ,T ] , P ) be a complete separable filtration probabilityspace. Let P be the predictable σ -field, that is the σ -field on [0 , T ] × Ω generated byall left continuous and F t -adapted real-valued processes. Further denote by BF the σ -field of the progressively measurable sets on [0 , T ] × Ω, i.e. BF = { O ⊂ [0 , T ] × Ω : ∀ t ∈ [0 , T ] , O ∩ ([0 , t ] × Ω) ∈ B ([0 , t ]) ⊗ F t } , where B ([0 , t ]) denotes the Borel σ -field on [0 , t ].Now we consider the following type of SPDEs driven by L´evy processes: dX ǫt = A ( t, X ǫt ) dt + ǫ Z X f ( t, X ǫt − , z ) e N ǫ − ( dt, dz ) , (1.1) X ǫ = x ∈ H, where A : [0 , T ] × V → V ∗ is a B ([0 , T ]) ⊗ B ( V )-measurable function. X is a locallycompact Polish space. N ǫ − is a Poisson random measure on [0 , T ] × X with a σ -finitemean measure ǫ − λ T ⊗ ν , λ T is the Lebesgue measure on [0 , T ] and ν is a σ -finitemeasure on X . e N ǫ − ([0 , t ] × B ) = N ǫ − ([0 , t ] × B ) − ǫ − tν ( B ) , ∀ B ∈ B ( X ) with ν ( B ) < ∞ , is the compensated Poisson random measure. f : [0 , T ] × V × X → H is a B ([0 , T ]) ⊗B ( V ) ⊗ B ( X )-measurable function.The following assumptions are from [6], which guarantee that Eq. (1.1) admits aunique solution. Suppose that there exists constants α > , β ≥ , θ > , C > K and F and a function ρ : V → [0 , + ∞ ) which is measurable andbounded on the balls, such that the following conditions hold for all v, v , v ∈ V and t ∈ [0 , T ]: (H1) (Hemicontinuity) The map s
7→ hA ( t, v + sv ) , v i V ∗ ,V is continuous on R . (H2) (Local monotonicity)2 hA ( t, v ) − A ( t, v ) , v − v i V ∗ ,V + Z X k f ( t, v , z ) − f ( t, v , z ) k H ν ( dz ) ≤ ( K t + ρ ( v )) k v − v k H , H3) (Coercivity) 2 hA ( t, v ) , v i V ∗ ,V + θ k v k αV ≤ F t (1 + k v k H ) . (H4) (Growth) kA ( t, v ) k αα − V ∗ ≤ ( F t + C k v k αV )(1 + k v k βH ) . Definition 1.1. An H -valued c´adl´ag F t -adapted process { X ǫt } t ∈ [0 ,T ] is called a solutionof Eq. (1.1) , if for its dt × P -equivalent class b X ǫ we have(1) b X ǫ ∈ L α ([0 , T ]; V ) ∩ L ([0 , T ]; H ) , P -a.s.;(2) the following equality holds P -a.s.: X ǫt = x + Z t A ( s, X ǫs ) ds + ǫ Z t Z X f ( s, X ǫs , z ) e N ǫ − ( ds, dz ) , t ∈ [0 , T ] , where X ǫ is any V -valued progressively measurable dt × P version of b X ǫ . With a minor modification of [6, Theorem 1.2], we have the following existence anduniqueness theorem for the solution of Eq. (1.1).
Theorem 1.2.
Suppose that conditions ( H - ( H hold for F, K ∈ L ([0 , T ]; R + ) , andthere exists a constant γ < θ β and G ∈ L ([0 , T ]; R + ) such that for all t ∈ [0 , T ] and v ∈ V we have Z X k f ( t, v, z ) k H ν ( dz ) ≤ F t (1 + k v k H ) + γ k v k αV ; (1.2) Z X k f ( t, v, z ) k β +2 H ν ( dz ) ≤ G t (1 + k v k β +2 H ); (1.3) ρ ( v ) ≤ C (1 + k v k αV )(1 + k v k βH ) . (1.4) Then(1) For any x ∈ L β +2 (Ω , F , P ; H ) , (1.1) has a unique solution { X ǫt } t ∈ [0 ,T ] .(2) If γ is small enough, then E (cid:16) sup t ∈ [0 ,T ] k X ǫt k β +2 H (cid:17) + E Z T k X ǫt k βH k X ǫt k αV dt ≤ C ǫ (cid:16) E k x k β +2 H + Z T G t dt + Z T F t dt (cid:17) . ǫ → D ([0 , T ] , H ), the space of H -valued c`adl`ag functions on [0 , T ].In the past three decades, there are numerous literatures about the LDP forstochastic evolution equations (SEEs) and SPDEs driven by Gaussian processes (cf.[5, 8, 9, 11, 12, 13, 14, 16, 21, 22, 24, 26, 28, 33, 34], etc.). Many of these resultswere obtained by using the weak convergence approach for the case of Gaussian noise,introduced by [8, 9], see, for example, [5, 8, 9, 16, 21, 22, 24, 26, 34]. This approachhas been proved to be very effective for various finite/infinite-dimensional stochasticdynamical systems. One of the main advantages of this approach is that one onlyneeds to make some necessary moment estimates.The situations for SEEs and SPDEs driven by L´evy noise are drastically differentbecause of the appearance of the jumps. There are only a few results on this topic sofar. The first paper on LDP for SEEs of jump type is R¨okner and Zhang [25] wherethe additive noise is considered. The study of LDP for multiplicative L´evy noise hasbeen carried out as well, e.g., [27] and [7] for SEEs where the LDP was establishedon a larger space (hence, with a weaker topology) than the actual state space of thesolution, [31] for SEEs on the actual state space, [32] for the 2-D stochastic Navier-Stokes equations (SNSEs). Before [32], Xu and Zhang[30] dealt with the 2-D SNSEsdriven by additive L´evy noise. We also refer to [1, 2, 4, 18] for related results.To obtain our result, we will use the weak convergence approach introduced by[7, 10, 17] for the case of Poisson random measures. This approach is a powerfultool to prove the LDP for SEEs and SPDEs driven by L´evy noise, which has beenapplied for several dynamical systems. The weak convergence method was first usedin [7] to obtain LDP for SPDEs on co-nuclear spaces driven by L ´ evy noises and in[31] for SPDEs on Hilbert spaces with regular coefficients. Paper [32] deals with the2-D SNSEs driven by multiplicative L´evy noise. Bao and Yuan [4] established a LDPfor a class of stochastic functional differential equations of neutral type driven by afinite-dimensional Wiener process and a stationary Poisson random measure.Monotone method is a main tool to prove the existence and uniqueness of SPDEs,and it can tackle a large class of SPDEs, for more details, see [6, 23] and referencestherein. Working in the framework of [6], the purpose of this paper is to establish a LDPfor a family of locally monotone SPDEs (1.1) driven by pure jumps. In addition to thedifficulties caused by the jumps, much of our problem is to deal with the monotoneoperator A . Using the weak convergence approach, the main point is to prove the4ightness of some controlled SPDEs, see (4.4). This is highly nontrivial. We first dividethe controlled SPDEs (4.4) into three parts, and establish the tightness of each partin suitable larger space, respectively, see Proposition 4.1. And then via the Skorohodrepresentation theorem we are able to show the weak convergence actually takes placein the spae D ([0 , T ] , H ). Finally, we mention that our framework can tackle the SPDEswith some polynomial growth, see Example 4.3 in [6].This paper is organized as follows. In Section 2, we will recall the abstract criteriafor LDP obtained in [7, 10]. In Section 3, we will show the main result of this paper.Section 4 and Section 5 is devoted to prove prior results on the controlled SPDEs (4.4),which play a key role in this paper. The entire Section 6 is to establish the LDP for(1.1). For convenience of the reader, we shall adopt the notation in [7] and [10]. Recall that X is a locally compact Polish space. Denote by M F C ( X ) the collection of all measureson ( X , B ( X )) such that ν ( K ) < ∞ for any compact K ∈ B ( X ) } . Denote by C c ( X )the space of continuous functions with compact supports, endow M F C ( X ) with theweakest topology such that for every f ∈ C c ( X ), the function ν → h f, ν i = Z X f ( u ) dν ( u ) , is continuous for ν ∈ M F C ( X ). This topology can be metrized such that M F C ( X ) isa Polish space (see e.g. [10]).Fixing T ∈ (0 , ∞ ), we denote X T = [0 , T ] × X and ν T = λ T ⊗ ν with λ T beingLebesgue measure on [0 , T ] and ν ∈ M F C ( X ). Let n be a Poisson random measureon X T with intensity measure ν T , it is well-known [20] that n is an M F C ( X T ) valuedrandom variable such that(i) for each B ∈ B ( X T ) with ν T ( B ) < ∞ , n ( B ) is Poisson distributed with mean ν T ( B );(ii) for disjoint B , · · · , B k ∈ B ( X T ), n ( B ) , · · · , n ( B k ) are mutually independentrandom variables.For notational simplicity, we write from now on M = M F C ( X T ) , (2.1)5nd denote by P the probability measure induced by n on ( M , B ( M )). Under P , thecanonical map, N : M → M , N ( m ) . = m , is a Poisson random measure with intensitymeasure ν T . With applications to large deviations in mind, we also consider, for θ > P θ on ( M , B ( M )) under which N is a Poisson random measurewith intensity θν T . The corresponding expectation operators will be denoted by E and E θ , respectively.For further use, simply denote Y = X × [0 , ∞ ) , Y T = [0 , T ] × Y , ¯ M = M F C ( Y T ) . (2.2)Let ¯ P be the unique probability measure on ( ¯ M , B ( ¯ M )) under which the canonicalmap, ¯ N : ¯ M → ¯ M , ¯ N ( ¯ m ) . = ¯ m , is a Poisson random measure with intensity measure¯ ν T = λ T ⊗ ν ⊗ λ ∞ , with λ ∞ being Lebesgue measure on [0 , ∞ ). The correspondingexpectation operator will be denoted by ¯ E . Let F t . = σ { ¯ N ((0 , s ] × A ) : 0 ≤ s ≤ t, A ∈B ( Y ) } , and let ¯ F t denote the completion under ¯ P . We denote by ¯ P the predictable σ -field on [0 , T ] × ¯ M with the filtration { ¯ F t : 0 ≤ t ≤ T } on ( ¯ M , B ( ¯ M )). Let ¯ A be theclass of all ( ¯ P ⊗ B ( X )) / B ([0 , ∞ ))-measurable maps ϕ : X T × ¯ M → [0 , ∞ ). For ϕ ∈ ¯ A ,we shall suppress the argument ¯ m in ϕ ( s, x, ¯ m ) and simply write ϕ ( s, x ) = ϕ ( s, x, ¯ m ).Define a counting process N ϕ on X T by N ϕ ((0 , t ] × U ) = Z (0 ,t ] × U Z (0 , ∞ ) [0 ,ϕ ( s,x )] ( r ) ¯ N ( dsdxdr ) , t ∈ [0 , T ] , U ∈ B ( X ) . (2.3)The above N ϕ is called a controlled random measure, with ϕ selecting the intensityfor the points at location x and time s , in a possibly random but non-anticipatingway. When ϕ ( s, x, ¯ m ) ≡ θ ∈ (0 , ∞ ), we write N ϕ = N θ . Note that N θ has the samedistribution with respect to ¯ P as N has with respect to P θ .Define l : [0 , ∞ ) → [0 , ∞ ) by l ( r ) = r log r − r + 1 , r ∈ [0 , ∞ ) . For any ϕ ∈ ¯ A the quantity L T ( ϕ ) = Z X T l ( ϕ ( t, x, ω )) ν T ( dtdx ) (2.4)is well defined as a [0 , ∞ ]-valued random variable.6 .2 A general criterion for large deviation principle [10, The-orem 4.2] We first state the large deviation principle we are concerned with. Let { X ǫ , ǫ > } ≡{ X ǫ } be a family of random variables defined on a probability space (Ω , F , P ) andtaking values in a Polish space E . Denote the expectation with respect to P by E . Thetheory of large deviations is concerned with events A for which probability P ( X ǫ ∈ A )converges to zero exponentially fast as ǫ →
0. The exponential decay rate of suchprobabilities is typically expressed in terms of a ’rate function’ I defined as below. Definition 2.1. (Rate function)
A function I : E → [0 , ∞ ] is called a rate functionon E , if for each M < ∞ the level set { y ∈ E : I ( y ) ≤ M } is a compact subset of E .For A ∈ B ( E ) , we define I ( A ) . = inf y ∈ A I ( y ) . Definition 2.2. (Large deviation principle)
Let I be a rate function on E . Thesequence { X ǫ } is said to satisfy a large deviation principle (LDP) on E with ratefunction I if the following two conditions hold.a. LDP upper bound. For each closed subset F of E , lim sup ǫ → ǫ log P ( X ǫ ∈ F ) ≤ − I ( F ) . b. LDP lower bound. For each open subset G of E , lim inf ǫ → ǫ log P ( X ǫ ∈ G ) ≥ − I ( G ) . Next, we recall the general criterion for large deviation principles established in[10]. Let {G ǫ } ǫ> be a family of measurable maps from M to U , where M is introducedin (2.1) and U is a Polish space. We present below a sufficient condition for LDP ofthe family Z ǫ = G ǫ (cid:16) ǫN ǫ − (cid:17) , as ǫ → S N = n g : X T → [0 , ∞ ) : L T ( g ) ≤ N o , (2.5)a function g ∈ S N can be identified with a measure ν gT ∈ M , defined by ν gT ( A ) = Z A g ( s, x ) ν T ( dsdx ) , A ∈ B ( X T ) . This identification induces a topology on S N under which S N is a compact space, seethe Appendix of [7]. Throughout this paper we use this topology on S N . Denote S = ∪ ∞ N =1 S N and ¯ A N := { ϕ ∈ ¯ A and ϕ ( ω ) ∈ S N , ¯ P - a.s. } .7 ondition 2.1. There exists a measurable map G : M → U such that the followinghold.a). For all N ∈ N , let g n , g ∈ S N be such that g n → g as n → ∞ . Then G (cid:16) ν g n T (cid:17) → G (cid:16) ν gT (cid:17) in U . b). For all N ∈ N , let ϕ ǫ , ϕ ∈ ¯ A N be such that ϕ ǫ converges in distribution to ϕ as ǫ → . Then G ǫ (cid:16) ǫN ǫ − ϕ ǫ (cid:17) ⇒ G (cid:16) ν ϕT (cid:17) . In this paper, we use the symbol “ ⇒ ” to denote convergence in distribution.For φ ∈ U , define S φ = n g ∈ S : φ = G (cid:16) ν gT (cid:17)o . Let I : U → [0 , ∞ ] be defined by I ( φ ) = inf g ∈ S φ L T ( g ) , φ ∈ U . (2.6)By convention, I ( φ ) = ∞ if S φ = ∅ . The following criterion for LDP was establishedin Theorem 4.2 of [10]. Theorem 2.3.
For ǫ > , let Z ǫ be defined by Z ǫ = G ǫ (cid:16) ǫN ǫ − (cid:17) , and suppose thatCondition 2.1 holds. Then the family { Z ǫ } ǫ> satisfies a large deviation principle withthe rate function I defined by (2.6) . For applications, the following strengthened form of Theorem 2.3 is more useful andwas established in Theorem 2.4 of [7]. Let { K n ⊂ X , n = 1 , , · · · } be an increasingsequence of compact sets such that ∪ ∞ n =1 K n = X . For each n , let¯ A b,n = n ϕ ∈ ¯ A : for all ( t, ω ) ∈ [0 , T ] × ¯ M , n ≥ ϕ ( t, x, ω ) ≥ /n if x ∈ K n and ϕ ( t, x, ω ) = 1 if x ∈ K cn o , and let ¯ A b = ∪ ∞ n =1 ¯ A b,n . Define ˜ A N = ¯ A N ∩ n φ : φ ∈ ¯ A b o . Theorem 2.4. [7]
Suppose Condition 2.1 holds with ¯ A N therein replaced by ˜ A N . Thenthe conclusions of Theorem 2.3 continue to hold . (1.1) Assume that X = x ∈ H is deterministic. Let X ǫ be the H -valued solution to Eq.(1.1) with initial value x . In this section, we state the LDP on D ([0 , T ] , H ) for { X ǫ } under suitable assumptions. 8ake U = D ([0 , T ] , H ) in Condition 2.1 with the Skorokhod topology U S . We knowthat ( U , U S ) is a Polish space. For p >
0, define H p = n h : [0 , T ] × X → R + : ∃ δ > , s.t. ∀ Γ ∈ B ([0 , T ]) ⊗ B ( X ) with ν T (Γ) < ∞ , we have Z Γ exp( δh p ( t, y )) ν ( dy ) dt < ∞ o . Remark 1.
It is easy to check that H p ⊂ H p ′ for any p ′ ∈ (0 , p ) and n h : [0 , T ] × X → R + , sup ( t,y ) ∈ [0 ,T ] × X h ( t, y ) < ∞ o ⊂ H p , ∀ p > . To study LDP of Eq. (1.1), besides the assumptions (H1)-(H4), we further need (H5)
There exist η > p ≥ Υ with Υ := β ( α − α + η ) α ∨ α − α + η ) α ∨ ∨ ( β + 2), and L f ∈ L ( ν T ) ∩ L ( ν T ) ∩ L β +2 ( ν T ) ∩ L Υ ( ν T ) ∩ L Υ2 ( ν T ) ∩ H p such that k f ( t, v, z ) k H ≤ L f ( t, z )(1 + k v k H ) , ∀ ( t, v, z ) ∈ [0 , T ] × V × X . (H6) There exists G f ∈ L ( ν T ) ∩ H such that k f ( t, v , z ) − f ( t, v , z ) k H ≤ G f ( t, z ) k v − v k H , ∀ ( t, z ) ∈ [0 , T ] × X , v , v ∈ V. Remark 2.
It is easy to check that L ( ν T ) ∩ n h : [0 , T ] × X → R + , k h k ∞ < ∞ o ⊂ L ( ν T ) ∩ L ( ν T ) ∩ L β +2 ( ν T ) ∩ L Υ ( ν T ) ∩ L Υ2 ( ν T ) ∩H p , where k h k ∞ = sup ( t,y ) ∈ [0 ,T ] × X h ( t, y ) . It follows from Theorem 1.2 that, for every ǫ >
0, there exists a measurable map G ǫ : ¯ M → D ([0 , T ]; H ) such that, for any Poisson random measure n ǫ − on [0 , T ] × X with mean measure ǫ − λ T ⊗ ν given on some probability space, G ǫ ( ǫ n ǫ − ) is the uniquesolution X ǫ of (1 .
1) with e N ǫ − replaced by e n ǫ − , here e n ǫ − is the compensated Poissonrandom measure of n ǫ − .To state our main result, we need to introduce the map G . Recall S given in Section2.2. For g ∈ S , consider the following deterministic PDE (the skeleton equation): X ,gt = x + Z t A ( s, X ,gs ) ds + Z t f ( s, X ,gs , z )( g ( s, z ) − ν ( dz ) ds, in V ∗ .
9y Proposition 5.1 below, this equation has a unique solution X ,g ∈ C ([0 , T ] , H ) ∩ L α ([0 , T ] , V ). Define G ( ν gT ) := X ,g , ∀ g ∈ S. (3.1)Let I : U = D ([0 , T ] , H ) → [0 , ∞ ] be defined as in (2.6). The following is the mainresult of this paper. Theorem 3.1.
Assume that (H1) - (H6) and (1.4) hold. Then the family { X ǫ } ǫ> satisfies an LDP on D ([0 , T ] , H ) with the rate function I under the topology of uniformconvergence.Proof. According to Theorem 2.4, we only need to verify Condition 2.1, which will bedone in the last section. G ǫ ( ǫN ǫ − ϕ ǫ ) In this section, we first state three lemmas whose proofs can be adopted from thosein [7], [31] and [10]. Then, we establish two key estimates for the stochastic processesstudied in this paper. Finally, we prove the tightness of this family of these stochasticprocesses.Using similar arguments as those in proving [7, Lemma 3.4], we can establish thefollowing lemma.
Lemma 4.1.
For any h ∈ H p ∩ L p ′ ( ν T ) , p ′ ∈ (0 , p ] , there exists a constant C h,p,p ′ ,N such that C h,p,p ′ ,N := sup g ∈ S N Z X T h p ′ ( s, v )( g ( s, v ) + 1) ν ( dv ) ds < ∞ . (4.1) For any h ∈ H ∩ L ( ν T ) , there exists a constant C h,N such that C h,N := sup g ∈ S N Z X T h ( s, v ) | g ( s, v ) − | ν ( dv ) ds < ∞ . (4.2)Using the argument used for proving [7, Lemmas 3.4 and 3.11] and [31, (3.19)], wefurther get Lemma 4.2.
Let h : X T → R be a measurable function such that Z X T | h ( s, v ) | ν ( dv ) ds < ∞ , nd for all δ ∈ (0 , ∞ ) Z E exp( δ | h ( s, v ) | ) ν ( dv ) ds < ∞ , for all E ∈ B ( X T ) satisfying ν T ( E ) < ∞ .a). Fix N ∈ N , and let g n , g ∈ S N be such that g n → g as n → ∞ . Then lim n →∞ Z X T h ( s, v )( g n ( s, v ) − ν ( dv ) ds = Z X T h ( s, v )( g ( s, v ) − ν ( dv ) ds ; b). Fix N ∈ N . Given ǫ > , there exists a compact set K ǫ ⊂ X , such that sup g ∈ S N Z [0 ,T ] Z K cǫ | h ( s, v ) || g ( s, v ) − | ν ( dv ) ds ≤ ǫ. c). For every η > , there exists δ > , we have such that for any A ∈ B ([0 , T ]) satisfying λ T ( A ) < δ sup g ∈ S N Z A Z X h ( s, v ) | g ( s, v ) − | ν ( dv ) ds ≤ η. (4.3)Fix N ∈ N . For any ϕ ǫ ∈ ˜ A N , consider the following controlled SPDEs d e X ǫt = A ( t, e X ǫt ) dt + Z X f ( t, e X ǫt , z )( ϕ ǫ ( t, z ) − ν ( dz ) dt + ǫ Z X f ( t, e X ǫt − , z ) e N ǫ − ϕ ǫ ( dz, dt ) , (4.4)with initial condition e X ǫ = x .Recall ˜ A N in Theorem 2.4. Let ϑ ǫ = ϕ ǫ . The following lemma follows from Lemma2.3 and Section 5.2 in [10]. Recall the notations in Section 2.1, we have Lemma 4.3. E ǫt ( ϑ ǫ ) := exp n Z (0 ,t ] × X × [0 ,ǫ − ϕ ǫ ] log( ϑ ǫ ( s, x )) ¯ N ( ds dx dr )+ Z (0 ,t ] × X × [0 ,ǫ − ϕ ǫ ] ( − ϑ ǫ ( s, x ) + 1)¯ ν T ( ds dx dr ) o Consequently, Q ǫt ( G ) = Z G E ǫt ( ϑ ǫ ) d ¯ P , for G ∈ B ( ¯ M ) defines a probability measure on ¯ M .
11y the fact that ǫN ǫ − ϕ ǫ under Q ǫT has the same law as that of ǫN ǫ − under ¯ P .From Theorem 1.2, we see that there exists a unique solution e X ǫ to the controlledSPDE (4.4) which satisfies (2) in Theorem 1.2.By the definition of G ǫ , we have e X ǫ = G ǫ (cid:16) ǫN ǫ − ϕ ǫ (cid:17) . (4.5)The following estimates (Lemmas 4.4 and 4.5) will be useful. Lemma 4.4.
For p = 2 , β or Υ in (H5), there exists ǫ p , C p > such that sup ǫ ∈ (0 ,ǫ p ] E (cid:16) sup t ∈ [0 ,T ] k e X ǫt k pH (cid:17) + E (cid:16) Z T k e X ǫt k p − H k e X ǫt k αV dt (cid:17) ≤ C p . Proof.
By Itˆo ′ s formula, we have k e X ǫt k pH = k x k pH + I ( t ) + I ( t ) + I ( t ) + I ( t ) , (4.6)where I ( t ) = p Z t k e X ǫs k p − H (cid:16) hA ( s, e X ǫs ) , e X ǫs i V ∗ ,V (cid:17) ds,I ( t ) = Z t Z X p k e X ǫs − k p − H h ǫf ( s, e X ǫs − , z ) , e X ǫs − i H,H e N ǫ − ϕ ǫ ( dz, ds ) ,I ( t ) = Z t Z X h k e X ǫs − + ǫf ( s, e X ǫs − , z ) k pH − k e X ǫs − k pH − p k e X ǫs − k p − H h ǫf ( s, e X ǫs − , z ) , e X ǫs − i H,H i N ǫ − ϕ ǫ ( dz, ds ) , and I ( t ) = p Z t k e X ǫs k p − H h Z X f ( s, e X ǫs , z )( ϕ ǫ ( s, z ) − , e X ǫs i H,H ν ( dz ) ds. Note that by (H3), I ( t ) ≤ p Z t k e X ǫs k p − H (cid:16) F s + F s k e X ǫs k H − θ k e X ǫs k αV (cid:17) ds ≤ − θp Z t k e X ǫs k p − H k e X ǫs k αV ds + p Z t h(cid:16) k e X ǫs k pH + 1 (cid:17) F s + F s k e X ǫs k pH i ds ≤ − θp Z t k e X ǫs k p − H k e X ǫs k αV ds + p Z t F s ds + Z t pF s k e X ǫs k pH ds, (4.7)12nd by (H5), I ( t ) ≤ p Z t k e X ǫs k p − H Z X k f ( s, e X ǫs , z ) k H | ( ϕ ǫ ( s, z ) − | ν ( dz ) ds ≤ p Z t k e X ǫs k p − H (1 + k e X ǫs k H ) Z X L f ( s, z ) | ( ϕ ǫ ( s, z ) − | ν ( dz ) ds ≤ p Z t Z X L f ( s, z ) | ( ϕ ǫ ( s, z ) − | ν ( dz ) ds +2 p Z t k e X ǫs k pH Z X L f ( s, z ) | ( ϕ ǫ ( s, z ) − | ν ( dz ) ds. (4.8)By Gronwall’s inequality, combining (4.6) (4.7), (4.8) and Lemma 4.1, k e X ǫt k pH + θp Z t k e X ǫs k p − H k e X ǫs k αV ds (4.9) ≤ exp (cid:16) p Z T F s ds + 2 pC L f ,N (cid:17) × (cid:16) k x k pH + p Z T F s ds + sup s ∈ [0 ,t ] | I ( s ) | + pC L f ,N + Z t Z X c p (cid:16) k e X ǫs − k p − H k ǫf ( s, e X ǫs − , z ) k H + k ǫf ( s, e X ǫs − , z ) k pH (cid:17) N ǫ − ϕ ǫ ( dz, ds ) (cid:17) , we have used (4.9) in [6] to I , i.e. (cid:12)(cid:12)(cid:12) k x + h k pH − k x k pH − p k x k p − H h x, h i H,H (cid:12)(cid:12)(cid:12) ≤ c p (cid:16) k x k p − H k h k H + k h k pH (cid:17) , ∀ x, h ∈ H. By Lemma 4.1, we have E (cid:16) sup s ∈ [0 ,T ] | I ( s ) | (cid:17) ≤ E (cid:16) Z T Z X ǫ p k e X ǫ ( s − ) k p − H h f ( s, e X ǫs − , z ) , e X ǫ ( s − ) i H,H N ǫ − ϕ ǫ ( dz, ds ) (cid:17) / ≤ E (cid:16) Z T Z X ǫ p k e X ǫ ( s − ) k p − H L f ( s, z ) (cid:16) k e X ǫs − k H + 1 (cid:17) N ǫ − ϕ ǫ ( dz, ds ) (cid:17) / ≤ E (cid:16) sup s ∈ [0 ,T ] k e X ǫs k pH · ǫ p Z T Z X k e X ǫs − k p − H L f ( s, z ) (cid:16) k e X ǫs − k H + 1 (cid:17) N ǫ − ϕ ǫ ( dz, ds ) (cid:17) / ≤ E (cid:16) sup s ∈ [0 ,T ] k e X ǫs k pH (cid:17) +16 ǫp E h(cid:16) sup s ∈ [0 ,T ] k e X ǫs k pH + 1 (cid:17) Z T Z X L f ( s, z ) ϕ ǫ ( s, z ) ν ( dz ) ds i ≤ ( 14 + 16 ǫp C L f , , ,N ) E (cid:16) sup s ∈ [0 ,T ] k e X ǫs k pH (cid:17) + 16 ǫp C L f , , ,N . (4.10)13n the other hand, by Lemma 4.1 again, we have E (cid:16) Z T Z X c p k e X ǫs k p − H k ǫf ( s, e X ǫs , z ) k H N ǫ − ϕ ǫ ( dz, ds ) (cid:17) ≤ ǫc p E (cid:16) Z T Z X k e X ǫs k p − H L f ( s, z )( k e X ǫs k H + 1) ϕ ǫ ( s, z ) ν ( dz ) ds (cid:17) ≤ ǫc p E h(cid:16) sup s ∈ [0 ,T ] k e X ǫs k pH + 1 (cid:17) Z T Z X L f ( s, z ) ϕ ǫ ( s, z ) ν ( dz ) ds i ≤ ǫc p C L f ,p,p,N E (cid:16) sup s ∈ [0 ,T ] k e X ǫs k pH (cid:17) + ǫc p C L f , , ,N , (4.11)and E (cid:16) Z T Z X c p k ǫf ( s, e X ǫs , z ) k pH N ǫ − ϕ ǫ ( dz, ds ) (cid:17) = ǫ p − c p E (cid:16) Z T Z X k f ( s, e X ǫs , z ) k pH ϕ ǫ ( s, z ) ν ( dz ) ds (cid:17) ≤ ǫ p − c p E h(cid:16) sup s ∈ [0 ,T ] k e X ǫs k pH + 1 (cid:17) Z T Z X L pf ( s, z ) ϕ ǫ ( s, z ) ν ( dz ) ds i ≤ ǫ p − c p C L f ,p,p,N E (cid:16) sup s ∈ [0 ,T ] k e X ǫs k pH (cid:17) + ǫ p − c p C L f ,p,p,N . (4.12)Combining (4.9)–(4.12), we obtain that there exists ǫ p > ǫ ∈ (0 ,ǫ p ] h E (cid:16) sup s ∈ [0 ,T ] k e X ǫs k pH (cid:17) + θp E (cid:16) Z T k e X ǫs k p − H k e X ǫs k αV ds (cid:17)i ≤ C N,p,T, k x k H , R T F s ds,L f . The proof is complete.
Lemma 4.5.
For p = Υ2 , there exist C p such that sup ǫ ∈ (0 ,ǫ p ] E (cid:16) Z T k e X ǫs k αV ds (cid:17) p ≤ C p . Here ǫ p comes from Lemma 4.4.Proof. Consider p = 2 in (4.9), we have θ Z t k e X ǫs k αV ds ≤ C N,T, R T F s ds,L f k x k H + Z T F s ds + sup s ∈ [0 ,t ] | I ( s ) | + 2 C L f ,N + J ( t ) ! , (4.13)where J ( t ) = Z t Z X c (cid:16) k ǫf ( s, e X ǫs − , z ) k H (cid:17) N ǫ − ϕ ǫ ( dz, ds ) .
14n the following calculations, we take p = Υ2 . Note that E ( | J ( t ) | p ) ≤ c p E (cid:16)(cid:12)(cid:12)(cid:12) Z T Z X (cid:16) k ǫf ( s, e X ǫs − , z ) k H (cid:17) e N ǫ − ϕ ǫ ( dz, ds ) (cid:12)(cid:12)(cid:12) p (cid:17) + c p E (cid:16)(cid:12)(cid:12)(cid:12) Z T Z X (cid:16) ǫ k f ( s, e X ǫs , z ) k H (cid:17) ϕ ǫ ( s, z ) ν ( dz ) ds (cid:12)(cid:12)(cid:12) p (cid:17) . By Kunita’s first inequality (refer to Theorem 4.4.23 in [3]), we can continue with E ( | J ( t ) | p ) ≤ c p ǫ p − E (cid:16) Z T Z X k f ( s, e X ǫs , z ) k pH ϕ ǫ ( s, z ) ν ( dz ) ds (cid:17) + c p ǫ p/ E (cid:16) Z T Z X k f ( s, e X ǫs , z ) k H ϕ ǫ ( s, z ) ν ( dz ) ds (cid:17) p/ + c p ǫ p E (cid:16) Z T Z X k f ( s, e X ǫs , z ) k H (cid:17) ϕ ǫ ( s, z ) ν ( dz ) ds (cid:17) p . Thus, by Lemma 4.1, we have E ( | J ( t ) | p ) (4.14) ≤ c p E (cid:16) s ∈ [0 ,T ] k e X ǫs k H (cid:17) p (cid:16) ǫ p − sup ϕ ∈ S N Z T Z X L pf ( s, z ) ϕ ( s, z ) ν ( dz ) ds + ǫ p/ (cid:16) sup ϕ ∈ S N Z T Z X L f ( s, z ) ϕ ( s, z ) ν ( dz ) ds (cid:17) p/ + ǫ p (cid:16) sup ϕ ∈ S N Z T Z X L f ( s, z ) ϕ ( s, z ) ν ( dz ) ds (cid:17) p (cid:17) ≤ c p E (cid:16) s ∈ [0 ,T ] k e X ǫs k (cid:17) p (cid:16) ǫ p − C L f , p, p,N + ǫ p/ (cid:16) C L f , , ,N (cid:17) p/ + ǫ p (cid:16) C L f , , ,N (cid:17) p (cid:17) .
15y Kunita’s first inequality again, E sup s ∈ [0 ,T ] | I ( s ) | p ! ≤ c p ǫ p − E (cid:16) Z T Z X (cid:12)(cid:12)(cid:12) h f ( s, e X ǫs , z ) , e X ǫs i H,H (cid:12)(cid:12)(cid:12) p ϕ ǫ ( s, z ) ν ( dz ) ds (cid:17) + c p ǫ p/ E (cid:16) Z T Z X (cid:12)(cid:12)(cid:12) h f ( s, e X ǫs , z ) , e X ǫs i H,H (cid:12)(cid:12)(cid:12) ϕ ǫ ( s, z ) ν ( dz ) ds (cid:17) p/ ≤ c p ǫ p − E (cid:16) Z T Z X k e X ǫs k pH L pf ( s, z ) (cid:16) k e X ǫs k H (cid:17) p ϕ ǫ ( s, z ) ν ( dz ) ds (cid:17) + c p ǫ p/ E (cid:16) Z T Z X k e X ǫs k H L f ( s, z ) (cid:16) k e X ǫs k H (cid:17) ϕ ǫ ( s, z ) ν ( dz ) ds (cid:17) p/ ≤ c p ǫ p − E (cid:16) s ∈ [0 ,T ] k e X ǫs k H (cid:17) p sup ϕ ∈ S N Z T Z X L pf ( s, z ) ϕ ( s, z ) ν ( dz ) ds + c p ǫ p/ E (cid:16) s ∈ [0 ,T ] k e X ǫs k H (cid:17) p (cid:16) sup ϕ ∈ S N Z T Z X L f ( s, z ) ϕ ( s, z ) ν ( dz ) ds (cid:17) p/ ≤ c p E (cid:16) s ∈ [0 ,T ] k e X ǫs k (cid:17) p (cid:16) ǫ p − C L f ,p,p,N + ǫ p/ (cid:16) C L f , , ,N (cid:17) p/ (cid:17) . (4.15)Lemma 4.4 and (4.13)–(4.15) imply this lemma.Finally, we prove the tightness of { e X ǫ } . Proposition 4.1.
For some ǫ > , { e X ǫ } ǫ ∈ (0 ,ǫ ] is tight in D ([0 , T ] , V ∗ ) with theSkorohod topology. Moreover, set M ǫt = Z t Z X ǫf ( s, e X ǫs − , z ) e N ǫ − ϕ ǫ ( dz, ds ) ,Z ǫt = Z t Z X f ( s, e X ǫs , z )( ϕ ǫ ( s, z ) − ν ( dz ) ds,Y ǫt = Z t A ( s, e X ǫs ) ds, then(a) lim ǫ → E (cid:16) sup t ∈ [0 ,T ] (cid:13)(cid:13)(cid:13) M ǫt (cid:13)(cid:13)(cid:13) H (cid:17) = 0 ,(b) ( Z ǫt ) ≤ t ≤ T is tight in C ([0 , T ] , V ∗ ) , c) ( Y ǫt ) ≤ t ≤ T is tight in C ([0 , T ] , V ∗ ) .Proof. (a). By Lemma 4.1, we have E (cid:16) sup t ∈ [0 ,T ] (cid:13)(cid:13)(cid:13) M ǫt (cid:13)(cid:13)(cid:13) H (cid:17) ≤ Cǫ E (cid:16) Z T Z X k f ( s, e X ǫs , z ) k H (cid:17) ϕ ǫ ( s, z ) ν ( dz ) ds (cid:17) ≤ Cǫ E (cid:16) Z T Z X L f ( s, z ) (cid:16) k e X ǫs k H (cid:17) ϕ ǫ ( s, z ) ν ( dz ) ds (cid:17) ≤ Cǫ E (cid:16) s ∈ [0 ,T ] k e X ǫs k H (cid:17) (cid:16) sup ϕ ∈ S N Z T Z X L f ( s, z ) ϕ ( s, z ) ν ( dz ) ds (cid:17) ≤ Cǫ E (cid:16) s ∈ [0 ,T ] k e X ǫs k H (cid:17) C L f , , ,N → , as ǫ → . (4.16)(b). It is sufficient to prove that for any δ >
0, there exists a compact subset K δ ⊂ C ([0 , T ] , V ∗ ) such that P ( Z ǫ ∈ K δ ) > − δ. Denote D M,N = n ( r t , g t ) : r · ∈ D ([0 , T ] , H ) ∩ L α ([0 , T ] , V ) , sup t ∈ [0 ,T ] k r t k H ≤ M ; g ∈ S N o , R ( D M,N ) = n y · = Z · Z X f ( s, r s , z )( g ( s, z ) − ν ( dz ) ds, ( r, g ) ∈ D M,N o . For any y ∈ R ( D M,N ), we have k y t − y s k H ≤ Z ts Z X k f ( l, r ( l ) , z ) k H | g ( l, z ) − | ν ( dz ) dl ≤ sup l ∈ [ s,t ] (1 + k r ( l ) k H ) Z ts Z X L f ( l, z ) | g ( l, z ) − | ν ( dz ) dl ≤ ( M + 1) sup ϕ ∈ S N Z ts Z X L f ( l, z ) | ϕ ( l, z ) − | ν ( dz ) dl. (4.17)Applying Lemma 4.1, c) in Lemma 4.2 and (4.17), we obtain the following:(1) for any η >
0, there exists ̟ > y ) such that for any s, t ∈ [0 , T ] and | t − s | ≤ ̟ k y t − y s k H ≤ η, ∀ y ∈ R ( D M,N ) , y ∈R ( D M,N ) sup t ∈ [0 ,T ] k y t k H = sup y ∈R ( D M,N ) sup t ∈ [0 ,T ] k y t − y k H ≤ ( M + 1) C L f ,N . Since
V ֒ → H is compact, we also have H ֒ → V ∗ compactly. By Ascoli-Arzel´a’stheorem, the complement of R ( D M,N ) in C ([0 , T ] , V ∗ ), denoted by R ( D M,N ), is acompact subset in C ([0 , T ] , V ∗ ).On the other hand, P ( Z ǫ ∈ R ( D M,N )) ≥ P ( sup t ∈ [0 ,T ] k e X ǫt k H ≤ M )= 1 − P ( sup t ∈ [0 ,T ] k e X ǫt k H > M ) ≥ − E ( sup t ∈ [0 ,T ] k e X ǫt k H ) /M ≥ − C /M , we have applied Lemma 4.4 in the last inequality and this establishes that { Z ǫ } istight in C ([0 , T ] , V ∗ ).(c). By Lemmas 4.4 and 4.5, recall η in (H5), let p = α + η , we have E k Y ǫt − Y ǫs k pV ∗ ≤ E (cid:12)(cid:12)(cid:12) Z ts kA ( l, e X ǫl ) k V ∗ dl (cid:12)(cid:12)(cid:12) p ≤ | t − s | p/α E (cid:16) Z ts kA ( l, e X ǫl ) k αα − V ∗ dl (cid:17) ( α − pα ≤ | t − s | p/α E (cid:16) Z ts ( F l + C k e X ǫl k αV )(1 + k e X ǫl k βH ) dl (cid:17) ( α − pα ≤ | t − s | p/α h E (cid:16) sup l ∈ [0 ,T ] (1 + k e X ǫl k βH ) α − pα (cid:17) + E (cid:16) Z ts F l + C k e X ǫl k αV dl (cid:17) α − pα i ≤ C α,p,F | t − s | p/α . Hence, a direct application of Kolmogorov’s criterion, for every ̟ ∈ (0 , α − p ),there exists constant C ̟ independent on ǫ such that E (cid:16) sup t = s ∈ [0 ,T ] k Y ǫt − Y ǫs k pV ∗ | t − s | p̟ (cid:17) ≤ C ̟ . (4.18)On the other hand, by (4.4), we have e X ǫt = x + Y ǫt + Z ǫt + M ǫt . E (cid:16) sup t ∈ [0 ,T ] k Y ǫt k H (cid:17) (4.19) ≤ C h k x k H + E (cid:16) sup t ∈ [0 ,T ] k e X ǫt k H (cid:17) + E (cid:16) sup t ∈ [0 ,T ] k Z ǫt k H (cid:17) + E (cid:16) sup t ∈ [0 ,T ] k M ǫt k H (cid:17)i . Notice that E (cid:16) sup t ∈ [0 ,T ] k Z ǫt k H (cid:17) ≤ E (cid:16) Z T Z X k f ( s, e X ǫs , z ) k H | ϕ ǫ ( s, z ) − | ν ( dz ) ds (cid:17) ≤ C E (cid:16) t ∈ [0 ,T ] k e X ǫt k H (cid:17) (cid:16) sup ϕ ∈ S N Z T Z X L f ( s, z ) | ϕ ( s, z ) − | ν ( dz ) ds (cid:17) ≤ CC L f ,N E (cid:16) t ∈ [0 ,T ] k e X ǫt k H (cid:17) (4.20)By Lemma 4.4, (4.19), (4.20) and (4.16), we have E (cid:16) sup t ∈ [0 ,T ] k Y ǫt k H (cid:17) ≤ C < ∞ , (4.21)where C is independent of ǫ .For ̟ ∈ (0 ,
1) and
R >
0. Set K R,̟ := n j ∈ C ([0 , T ] , V ∗ ) : sup t ∈ [0 ,T ] k j t k H + sup s = t ∈ [0 ,T ] k j t − j s k V ∗ | t − s | ̟ ≤ R o . Since
V ֒ → H is compact, we also have H ֒ → V ∗ compactly. By Ascoli-Arzel´a’stheorem, K R,̟ is a compact subset of C ([0 , T ] , V ∗ ). By (4.18), (4.21) and Chebyschev’sinequality, for some ̟ ∈ (0 ,
1) and any
R >
0, we have P (cid:16) Y ǫ K R,̟ (cid:17) ≥ C T,̟
R .
This implies the tightness of { Y ǫ } in C ([0 , T ] , V ∗ ).The tightness of { e X ǫ } in D ([0 , T ] , V ∗ ) then follows from (4.4) and the conclusionsproved above. With the tightness result obtained in the last section, we now characterize the limitpoints and derive limiting results for the processes.19hroughout this section, we assume that for almost all ω , as ǫ → ϕ ǫ ( · , · )( ω )converges to ϕ ( · , · )( ω ) in S N weakly, and X ǫ ( ω ) converges to X ( ω ) in D ([0 , T ] , V ∗ )strongly with supremum norm.Set K = L α ([0 , T ] × Ω → V ; dt × ¯ P ) , K ∗ = L αα − ([0 , T ] × Ω → V ∗ ; dt × ¯ P ) . Lemma 5.1.
There exists a subsequence ( ǫ k ) , ¯ X ∈ K ∩ L ∞ ([0 , T ] , L β +2 (Ω , H )) and Y ∈ K ∗ such that(i) X ǫ k → ¯ X in K weakly and in L ∞ ([0 , T ] , L β +2 (Ω , H )) in weak-star topology,(ii) A ( · , X ǫ k ) → Y in K ∗ weakly,(iii) lim ǫ → E (cid:16) sup t ∈ [0 ,T ] k X ǫt − X t k V ∗ (cid:17) = 0 , and for m = αα +1 , lim ǫ → E Z T k X ǫt − X t k mH dt = 0 . Proof. (i) following from Lemma 4.4. For (ii), by Lemma 4.4 again, kA ( · , X ǫ ( · )) k α − α K ∗ = E (cid:16) Z T kA ( t, X ǫt ) k αα − V ∗ dt (cid:17) ≤ E (cid:16) Z T ( F t + C k X ǫt k αV )(1 + k X ǫt k βH ) dt (cid:17) ≤ C < ∞ . (5.1)Lemma 4.4 implies E (cid:16) sup t ∈ [0 ,T ] k X ǫt k H (cid:17) ≤ C ,N,x , (5.2)and E (cid:16) Z T k X ǫt k αV dt (cid:17) ≤ C. (5.3)Hence, by the strong convergence of X ǫ ( ω ) to X ( ω ) in D ([0 , T ] , V ∗ ) with sup norm,Fatou’s lemma, (5.2) and (5.3), we have E (cid:16) sup t ∈ [0 ,T ] k X t k H (cid:17) ≤ lim inf ǫ → E (cid:16) sup t ∈ [0 ,T ] k X ǫt k H (cid:17) ≤ C ,N,x , (5.4)20 (cid:16) Z T k X t k αV dt (cid:17) ≤ lim inf ǫ → E (cid:16) Z T k X ǫt k αV dt (cid:17) ≤ C. (5.5)and lim ǫ → E (cid:16) sup t ∈ [0 ,T ] k X ǫt − X t k V ∗ (cid:17) = 0 . (5.6)(5.6) can be seen as following. SetΩ ǫδ = { ω : sup t ∈ [0 ,T ] k X ǫt − X t k V ∗ ≥ δ } . The strong convergence of X ǫ ( ω ) to X ( ω ) in D ([0 , T ] , V ∗ ) with sup norm implieslim ǫ → P (Ω ǫδ ) = 0 , ∀ δ > . (5.7)Applying (5.7), (5.2) and (5.4) to (5.6), we havelim ǫ → E (cid:16) sup t ∈ [0 ,T ] k X ǫt − X t k V ∗ (cid:17) = lim ǫ → h E (cid:16) sup t ∈ [0 ,T ] k X ǫt − X t k V ∗ · Ω ǫδ (cid:17) + E (cid:16) sup t ∈ [0 ,T ] k X ǫt − X t k V ∗ · (Ω ǫδ ) c (cid:17)i ≤ δ + lim ǫ → (cid:16) E (cid:16) sup t ∈ [0 ,T ] k X ǫt − X t k V ∗ (cid:17)(cid:17) / · (cid:16) P (Ω ǫδ ) (cid:17) / ≤ δ. The arbitrary of δ implies (5.6).Taking m = αα +1 , we get E Z T k X ǫt − X t k mH dt = E Z T h X ǫt − X t , X ǫt − X t i mV ∗ ,V dt ≤ E Z T k X ǫt − X t k mV ∗ k X ǫt − X t k mV dt ≤ (cid:16) E Z T k X ǫt − X t k V ∗ dt (cid:17) α − mα (cid:16) E Z T k X ǫt − X t k αV dt (cid:17) mα . Combining (5.3), (5.5) and (5.6), we havelim ǫ → E Z T k X ǫt − X t k mH dt = 0 . (5.8)21 emma 5.2. For any h ∈ H , we have lim ǫ k → h Z t Z X f ( s, X ǫ k s , z )( ϕ ǫ k ( s, z ) − ν ( dz ) ds, h i H,H = h Z t Z X f ( s, X s , z )( ϕ ( s, z ) − ν ( dz ) ds, h i H,H . (5.9) Proof.
Denote ζ ( s, z ) = h f ( s, X s , z ) , h i H,H . Since sup s ∈ [0 ,T ] k X s k H < ∞ , P -a.s., and L f ∈ H , it follows from Remark 1 and Lemma 4.2 thatlim ǫ k → h Z t Z X f ( s, X s , z )( ϕ ǫ k ( s, z ) − ν ( dz ) ds, h i H,H = h Z t Z X f ( s, X s , z )( ϕ ( s, z ) − ν ( dz ) ds, h i H,H . (5.10)For any δ >
0, denote A δ,ǫ ( ω ) := n s ∈ [0 , T ] : k X ǫs − X s k H > δ o . By (5.8)lim ǫ → E (cid:16) λ T ( A δ,ǫ ) (cid:17) ≤ δ m lim ǫ → E Z T k X ǫt − X t k mH dt = 0 . Therefore, there exists a subsequence ǫ k (for simplicity, we still denote it by the samenotation ǫ k ) such that lim ǫ k → λ T ( A δ,ǫ k ) = 0 , ¯ P -a.s. . (5.11)Applying Lemma 4.1, we have Z T Z X k f ( s, X ǫ k s , z ) − f ( s, X s , z ) k H | ϕ ǫ k ( s, z ) − | ν ( dz ) ds ≤ Z T Z X G f ( s, z ) k X ǫ k s − X s k H | ϕ ǫ k ( s, z ) − | ν ( dz ) ds ≤ δ Z A cδ,ǫk Z X G f ( s, z ) | ϕ ǫ k ( s, z ) − | ν ( dz ) ds + sup s ∈ [0 ,T ] k X ǫ k s − X s k H Z A δ,ǫk Z X G f ( s, z ) | ϕ ǫ k ( s, z ) − | ν ( dz ) ds ≤ δ sup ϕ ∈ S N Z T Z X G f ( s, z ) | ϕ ( s, z ) − | ν ( dz ) ds + sup s ∈ [0 ,T ] k X ǫ k s − X s k H sup ϕ ∈ S N Z A δ,ǫk Z X G f ( s, z ) | ϕ ( s, z ) − | ν ( dz ) ds (5.12) ≤ δC G f ,N + sup s ∈ [0 ,T ] k X ǫ k s − X s k H sup ϕ ∈ S N Z A δ,ǫk Z X G f ( s, z ) | ϕ ( s, z ) − | ν ( dz ) ds. E (cid:16) sup s ∈ [0 ,T ] k X ǫ k s − X s k H sup ϕ ∈ S N Z A δ,ǫk Z X G f ( s, z ) | ϕ ( s, z ) − | ν ( dz ) ds (cid:17) ≤ (cid:16) E (cid:16) sup s ∈ [0 ,T ] k X ǫ k s − X s k H (cid:17) (cid:16) E (cid:16) sup ϕ ∈ S N Z A δ,ǫk Z X G f ( s, z ) | ϕ ( s, z ) − | ν ( dz ) ds (cid:17) (cid:17) . (5.13)By the dominated convergence theorem, Lemma 4.2 c) and Lemma 4.1, we havelim ǫ k → E (cid:16) sup ϕ ∈ S N Z A δ,ǫk Z X G f ( s, z ) | ϕ ( s, z ) − | ν ( dz ) ds (cid:17) = 0 . (5.14)Hence, (5.2), (5.4), (5.12)-(5.14) implylim ǫ k → E (cid:16) Z T Z X k f ( s, X ǫ k s , z ) − f ( s, X s , z ) k H | ϕ ǫ k ( s, z ) − | ν ( dz ) ds (cid:17) = 0 . (5.15)So, there exists a subsequence ǫ k (for simplicity, we still denote it by the same notation ǫ k ) such thatlim ǫ k → Z T Z X k f ( s, X ǫ k s , z ) − f ( s, X s , z ) k H | ϕ ǫ k ( s, z ) − | ν ( dz ) ds = 0 , ¯ P - a.s.. Combining this with (5.10), we arrive at (5.9).Define e X t := x + Z t Y s ds + Z t Z X f ( s, X s , z )( ϕ ( s, z ) − ν ( dz ) ds. (5.16)By taking weak limit of (4.4), it is not difficulty to see that e X t ( ω ) = ¯ X t ( ω ) = X t ( ω ) , for dt × ¯ P -almost all ( t, ω ) . Set N := n φ : φ is a V -valued ¯ F t -adapted process such that E (cid:16) Z T ρ ( φ s ) ds (cid:17) < ∞ o . Fix φ ∈ K ∩ N ∩ L ∞ ([0 , T ] , L β +2 (Ω , H )) and ψ ∈ L ∞ ([0 , T ] , R ). Denote G ( X, ϕ, Y ) := E h Z T ψ t Z t e − R s ( K l + ρ ( φ l )) dl × (cid:28)Z X f ( s, X s , z )( ϕ ( s, z ) − ν ( dz ) , Y s (cid:29) H,H dsdt i . The following limiting result will be needed later.23 emma 5.3. lim ǫ k → G ( X ǫ k , ϕ ǫ k , X ǫ k ) = G ( X, ϕ, X ) . (5.17) Proof.
For any fixed ( t, ω ) ∈ [0 , T ] × Ω. Set ζ ( s, z ) = ψ t e − R s ( K l + ρ ( φ l )) dl h f ( s, X s , z ) , X s i H,H . By Lemma 4.2 and sup s ∈ [0 ,T ] k X s k H < ∞ ¯ P -a.s., we have ∀ ( t, ω ) ∈ [0 , T ] × Ω,lim ǫ k → ψ t Z t e − R s ( K l + ρ ( φ l )) dl (cid:28)Z X f ( s, X s , z )( ϕ ǫ k ( s, z ) − ν ( dz ) , X s (cid:29) H,H ds = ψ t Z t e − R s ( K l + ρ ( φ l )) dl (cid:28)Z X f ( s, X s , z )( ϕ ( s, z ) − ν ( dz ) , X s (cid:29) H,H ds.
On the other hand, by Lemma 4.1sup ϕ ∈ S N (cid:12)(cid:12)(cid:12) ψ t (cid:16) Z t e − R s ( K l + ρ ( φ l )) dl (cid:16) h Z X f ( s, X s , z )( ϕ ( s, z ) − ν ( dz ) , X s i H,H (cid:17) ds (cid:12)(cid:12)(cid:12) ≤ C ψ sup ϕ ∈ S N Z T Z X k f ( s, X s , z ) k H k X s k H | ϕ ( s, z ) − | ν ( dz ) ds ≤ C ψ (1 + sup s ∈ [0 ,T ] k X s k H ) sup ϕ ∈ S N Z T Z X L f ( s, z ) | ϕ ( s, z ) − | ν ( dz ) ds ≤ C ψ,L f ,N (1 + sup s ∈ [0 ,T ] k X s k H ) . By the dominated convergence theorem, we havelim ǫ k → G ( X, ϕ ǫ k , X ) = G ( X, ϕ, X ) . (5.18)Let δ >
0. Recall A δ,ǫ k := n s ∈ [0 , T ] : k X ǫ k s − X s k H > δ o , and (5.11) that is there exists a subsequence ǫ k such thatlim ǫ k → λ T ( A δ,ǫ k ) = 0 , P -a.s..24hen we have (cid:12)(cid:12)(cid:12) G ( X ǫ k , ϕ ǫ k , X ǫ k ) − G ( X ǫ k , ϕ ǫ k , X ) (cid:12)(cid:12)(cid:12) ≤ C E (cid:16) Z T Z X k f ( s, X ǫ k s , z ) k H | ϕ ǫ k ( s, z ) − |k X ǫ k s − X s k H ν ( dz ) ds (cid:17) ≤ C E (cid:16) Z T Z X L f ( s, z )(1 + k X ǫ k s k H ) | ϕ ǫ k ( s, z ) − |k X ǫ k s − X s k H ν ( dz ) ds (cid:17) ≤ Cδ E (cid:16) Z A cδ,ǫk Z X L f ( s, z )(1 + k X ǫ k s k H ) | ϕ ǫ k ( s, z ) − | ν ( dz ) ds (cid:17) + C E (cid:16) Z A δ,ǫk Z X L f ( s, z )(1 + k X ǫ k s k H ) | ϕ ǫ k ( s, z ) − |k X ǫ k s − X s k H ν ( dz ) ds (cid:17) ≤ Cδ E (cid:16) sup s ∈ [0 ,T ] (1 + k X ǫ k s k H ) (cid:17) sup ϕ ∈ S N Z T Z X L f ( s, z ) | ϕ ( s, z ) − | ν ( dz ) ds + C E h sup s ∈ [0 ,T ] (cid:16) (1 + k X ǫ k s k H )( k X ǫ k s − X s k H ) (cid:17) × sup ϕ ∈ S N Z A δ,ǫk Z X L f ( s, z ) | ϕ ( s, z ) − | ν ( dz ) ds i ≤ δC L f ,N + C (cid:16) E (cid:16) s ∈ [0 ,T ] k X ǫ k s k H (cid:17) (cid:17) / (cid:16) E (cid:16) s ∈ [0 ,T ] k X ǫ k s − X s k H (cid:17) (cid:17) / · (cid:16) E (cid:16) sup ϕ ∈ S N Z A δ,ǫk Z X L f ( s, z ) | ϕ ( s, z ) − | ν ( dz ) ds (cid:17) (cid:17) / . (5.19)Similar as (5.14) and (5.15), we havelim ǫ k → (cid:12)(cid:12)(cid:12) G ( X ǫ k , ϕ ǫ k , X ǫ k ) − G ( X ǫ k , ϕ ǫ k , X ) (cid:12)(cid:12)(cid:12) = 0 . (5.20)On the other hand, (cid:12)(cid:12)(cid:12) G ( X ǫ k , ϕ ǫ k , X ) − G ( X, ϕ ǫ k , X ) (cid:12)(cid:12)(cid:12) ≤ C E (cid:16) Z T Z X k f ( s, X ǫ k s , z ) − f ( s, X s , z ) k H | ϕ ǫ k ( s, z ) − |k X s k H ν ( dz ) ds (cid:17) ≤ C E (cid:16) Z T Z X G f ( s, z ) k X ǫ k s − X s k H | ϕ ǫ k ( s, z ) − |k X s k H ν ( dz ) ds (cid:17) . Using the similar arguments as proving (5.20), we havelim ǫ k → (cid:12)(cid:12)(cid:12) G ( X ǫ k , ϕ ǫ k , X ) − G ( X, ϕ ǫ k , X ) (cid:12)(cid:12)(cid:12) = 0 . (5.21)Combining (5.20), (5.21), and (5.18), we have (5.17).25 emma 5.4. Y t ( ω ) = A ( t, X t ( ω )) for dt × ¯ P -almost all ( t, ω ) . Proof.
For φ ∈ K ∩ N ∩ L ∞ ([0 , T ] , L β +2 (Ω , H )), applying the Itˆo ′ s formula, e − R t ( K s + ρ ( φ s )) ds k X ǫ k t k H − k x k H = Z t e − R s ( K l + ρ ( φ l )) dl h − ( K s + ρ ( φ s )) k X ǫ k s k H + 2 hA ( s, X ǫ k s ) , X ǫ k s i V ∗ ,V +2 h Z X f ( s, X ǫ k s , z )( ϕ ǫ k ( s, z ) − ν ( dz ) , X ǫ k s i H,H i ds + Z t e − R s ( K l + ρ ( φ l )) dl Z X h ǫ k h f ( s, X ǫ k s − , z ) , X ǫ k s − i H,H i ˜ N ǫ k − ϕ ǫk ( ds, dz )+ Z t e − R s ( K l + ρ ( φ l )) dl Z X h ǫ k k f ( s, X ǫ k s − , z ) k H i N ǫ k − ϕ ǫk ( ds, dz ) . Notice that M ǫ k ( t ) := Z t e − R s ( K l + ρ ( φ l )) dl Z X h ǫ k h f ( s, X ǫ k s − , z ) , X ǫ k s − i H,H i ˜ N ǫ k − ϕ ǫk ( ds, dz )is a square integrable martingale, we have E (cid:16) e − R t ( K s + ρ ( φ s )) ds k X ǫ k t k H (cid:17) − k x k H = − E (cid:16) Z t e − R s ( K l + ρ ( φ l )) dl ( K s + ρ ( φ s )) (cid:16) k X ǫ k s − φ s k H + 2 h X ǫ k s , φ s i H,H − k φ s k H (cid:17) ds (cid:17) + E (cid:16) Z t e − R s ( K l + ρ ( φ l )) dl (cid:16) hA ( s, X ǫ k s ) − A ( s, φ s ) , X ǫ k s − φ s i V ∗ ,V +2 hA ( s, φ s ) , X ǫ k s − φ s i V ∗ ,V + 2 hA ( s, X ǫ k s ) , φ s i V ∗ ,V (cid:17) ds (cid:17) + E (cid:16) Z t e − R s ( K l + ρ ( φ l )) dl (cid:16) h Z X f ( s, X ǫ k s , z )( ϕ ǫ k ( s, z ) − ν ( dz ) , X ǫ k s i H,H (cid:17) ds (cid:17) + E (cid:16) ǫ k Z t e − R s ( K l + ρ ( φ l )) dl Z X k f ( s, X ǫ k s , z ) k H ϕ ǫ k ( s, z ) ν ( dz ) ds (cid:17) ≤ − E (cid:16) Z t e − R s ( K l + ρ ( φ l )) dl ( K s + ρ ( φ s )) (cid:16) h X ǫ k s , φ s i H,H − k φ s k H (cid:17) ds (cid:17) + E (cid:16) Z t e − R s ( K l + ρ ( φ l )) dl (cid:16) hA ( s, φ s ) , X ǫ k s − φ s i V ∗ ,V + 2 hA ( s, X ǫ k s ) , φ s i V ∗ ,V (cid:17) ds (cid:17) + E (cid:16) Z t e − R s ( K l + ρ ( φ l )) dl (cid:16) h Z X f ( s, X ǫ k s , z )( ϕ ǫ k ( s, z ) − ν ( dz ) , X ǫ k s i H,H (cid:17) ds (cid:17) + E (cid:16) ǫ k Z t e − R s ( K l + ρ ( φ l )) dl Z X k f ( s, X ǫ k s , z ) k H ϕ ǫ k ( s, z ) ν ( dz ) ds (cid:17) . (5.22)26y (i) of Lemma 5.1, we get E h Z T ψ t (cid:16) e − R t ( K s + ρ ( φ s )) ds k X t k H − k x k H (cid:17) dt i ≤ lim inf ǫ k → E h Z T ψ t (cid:16) e − R t ( K s + ρ ( φ s )) ds k X ǫ k t k H − k x k H (cid:17) dt i . (5.23)By Lemma 4.1, E (cid:16) ǫ k Z t e − R s ( K l + ρ ( φ l )) dl Z X k f ( s, X ǫ k s , z ) k H ϕ ǫ k ( s, z ) ν ( dz ) ds (cid:17) ≤ E (cid:16) ǫ k Z t Z X (1 + k X ǫ k s k H ) L f ( s, z ) ϕ ǫ k ( s, z ) ν ( dz ) ds (cid:17) ≤ ǫ k E (cid:16) (1 + sup s ∈ [0 ,T ] k X ǫ k s k H ) (cid:17) sup ϕ ∈ S N Z T Z X L f ( s, z ) ϕ ( s, z ) ν ( dz ) ds ≤ ǫ k C L f , , ,N . (5.24)Combining from (5.22) to (5.24), and Lemma 5.3, we infer E h Z T ψ t (cid:16) e − R t ( K s + ρ ( φ s )) ds k X t k H − k x k H (cid:17) dt i (5.25) ≤ − E h Z T ψ t Z t e − R s ( K l + ρ ( φ l )) dl ( K s + ρ ( φ s )) (cid:16) h X s , φ s i H,H − k φ s k H (cid:17) dsdt i + E h Z T ψ t Z t e − R s ( K l + ρ ( φ l )) dl (cid:16) hA ( s, φ s ) , X s − φ s i V ∗ ,V + 2 h Y s , φ s i V ∗ ,V (cid:17) dsdt i + E h Z T ψ t Z t e − R s ( K l + ρ ( φ l )) dl (cid:16) h Z X f ( s, X s , z )( ϕ ( s, z ) − ν ( dz ) , X s i H,H (cid:17) dsdt i . On the other hand, by (5.16), we have E (cid:16) e − R t ( K s + ρ ( φ s )) ds k X t k H − k x k H (cid:17) (5.26)= − E (cid:16) Z t e − R s ( K l + ρ ( φ l )) dl ( K s + ρ ( φ s )) k X s k H ds (cid:17) + E (cid:16) Z t e − R s ( K l + ρ ( φ l )) dl h Y s , X s i V ∗ ,V ds (cid:17) + E (cid:16) Z t e − R s ( K l + ρ ( φ l )) dl (cid:16) h Z X f ( s, X s , z )( ϕ ( s, z ) − ν ( dz ) , X s i H,H (cid:17) ds (cid:17) . By (5.25) and (5.26), we have E h Z T ψ t Z t e − R s ( K l + ρ ( φ l )) dl (cid:16) − ( K s + ρ ( φ s )) k X s − φ s k H +2 hA ( s, φ s ) − Y s , X s − φ s i V ∗ ,V (cid:17)(cid:17) dsdt i ≤ . φ = X − η ˜ φv for ˜ φ ∈ L ∞ ([0 , T ] × Ω; dt × ¯ P ; R ) and v ∈ V , divide both sides by η and let η →
0, then we have E h Z T ψ t Z t e − R s ( K l + ρ ( φ l )) dl (cid:16) φ s hA ( s, φ s ) − Y s , v i V ∗ ,V (cid:17)(cid:17) dsdt i ≤ . Hence Y = A ( · , X ). Proposition 5.1. X ( ω ) solves the following equation: X t ( ω ) = x + Z t A ( s, X s ( ω )) ds + Z t Z X f ( s, X s ( ω ) , z )( ϕ ( s, z )( ω ) − ν ( dz ) ds, (5.27) which has an unique solution in C ([0 , T ] , H ) ∩ L α ([0 , T ] , V ) . Proof.
The equation (5.27) follows from Lemmas 5.1-5.4. The proof of the uniquenessis standard, and it is omitted.
Lemma 5.5.
There exists a subsequence ̟ k , such that lim ̟ k → sup t ∈ [0 ,T ] k X ̟ k t − X t k H = 0 , ¯ P -a.s. . (5.28) Proof.
Set L ǫ k t = X ǫ k t − X t . Then e − R t ( K s + ρ ( X s )) ds k L ǫ k t k H = Z t e − R s ( K r + ρ ( X r )) dr (cid:16) − ( K s + ρ ( X s )) k L ǫ k s k H +2 hA ( s, X ǫ k s ) − A ( s, X s ) , L ǫ k s i V ∗ ,V (cid:17) ds +2 Z t e − R s ( K r + ρ ( X r )) dr D Z X f ( s, X ǫ k s , z )( ϕ ǫ k ( s, z ) − ν ( dz ) − Z X f ( s, X s , z )( ϕ ( s, z ) − ν ( dz ) , L ǫ k s E H,H ds +2 ǫ k Z t e − R s ( K r + ρ ( X r )) dr h Z X f ( s, X ǫ k s , z ) , L ǫ k s i H,H ˜ N ǫ k − ϕ ǫk ( dz, ds )+ ǫ k Z t e − R s ( K r + ρ ( X r )) dr Z X k f ( s, X ǫ k s , z ) k H N ǫ k − ϕ ǫk ( dz, ds )= I ( t ) + I ( t ) + I ( t ) + I ( t ) . (5.29)(H2) implies I ( t ) ≤ . (5.30)28y (5.19) and (5.20), we have E (cid:16) sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12) Z t e − R s ( K r + ρ ( X r )) dr h Z X f ( s, X ǫ k s , z )( ϕ ǫ k ( s, z ) − ν ( dz ) , L ǫ k s i H,H ds (cid:12)(cid:12)(cid:12)(cid:17) ≤ E (cid:16) Z T Z X k f ( s, X ǫ k s , z ) k H | ϕ ǫ k ( s, z ) − |k L ǫ k s k H ν ( dz ) ds (cid:17) ≤ E (cid:16) Z T Z X k X ǫ k s k H L f ( s, z ) | ϕ ǫ k ( s, z ) − |k L ǫ k s k H ν ( dz ) ds (cid:17) → , as ǫ k → . (5.31)Then it is not difficulty to obtainlim ǫ k → E (cid:16) sup t ∈ [0 ,T ] | I ( t ) | (cid:17) = 0 . (5.32)For I , E (cid:16) sup t ∈ [0 ,T ] | I ( t ) | (cid:17) ≤ E (cid:16) Z T Z X ǫ k k L ǫ k s k H k f ( s, X ǫ k s , z ) k H N ǫ k − ϕ ǫk ( ds, dz ) (cid:17) / ≤ E (cid:16) √ ǫ k sup s ∈ [0 ,T ] k L ǫ k s k H (cid:16) Z T Z X ǫ k k f ( s, X ǫ k s , z ) k H N ǫ k − ϕ ǫk ( ds, dz ) (cid:17) / (cid:17) ≤ √ ǫ k (cid:16) E (cid:16) sup t ∈ [0 ,T ] k L ǫ k t k H (cid:17)(cid:17) / (cid:16) E (cid:16) Z T Z X k f ( s, X ǫ k s , z ) k H ϕ ǫ k ( s, z ) ν ( dz ) ds (cid:17)(cid:17) / ≤ √ ǫ k (cid:16) E (cid:16) sup t ∈ [0 ,T ] k L ǫ k t k H (cid:17)(cid:17) / × (cid:16) E (cid:16) t ∈ [0 ,T ] k X ǫ k t k H (cid:17) sup ϕ ∈ S N Z T Z X L f ( s, z ) ϕ ( s, z ) ν ( dz ) ds (cid:17) / → , as ǫ k → . (5.33)For I , E (cid:16) sup t ∈ [0 ,T ] | I ( t ) | (cid:17) ≤ ǫ k E (cid:16) Z T Z X k f ( s, X ǫ k s , z ) k H ϕ ǫ k ( s, z ) ν ( dz ) ds (cid:17) ≤ ǫ k E (cid:16) t ∈ [0 ,T ] k X ǫ k t k H (cid:17) sup ϕ ∈ S N Z T Z X L f ( s, z ) ϕ ( s, z ) ν ( dz ) ds → , as ǫ k → . (5.34)Combining (5.29)–(5.34), we havelim ǫ k → E (cid:16) sup t ∈ [0 ,T ] (cid:16) e − R t ( K s + ρ ( X s )) ds k L ǫ k t k H (cid:17)(cid:17) = 0 . ǫ k → E (cid:16) e − R T ( K s + ρ ( X s )) ds (cid:16) sup t ∈ [0 ,T ] k L ǫ k t k H (cid:17)(cid:17) = 0 . This implies that there exists a subsequence ̟ k such that X ̟ k converges to X ¯ P -a.s.. Recall (4.5) and (3.1), we have
Theorem 6.1.
Fixed N ∈ N , and let ϕ ǫ , ϕ ∈ ˜ A N be such that ϕ ǫ converges in distri-bution to ϕ as ǫ → . Then G ǫ ( ǫN ǫ − ϕ ǫ ) ⇒ G ( ν ϕT ) . Proof.
Recall ¯ M in Section 2 and notations in Proposition 4.1. DenoteΠ = (cid:16) S N , D ([0 , T ] , V ∗ ) , C ([0 , T ] , V ∗ ) , C ([0 , T ] , V ∗ ) , ¯ M (cid:17) . Proposition 4.1 implies that the laws of n(cid:16) ϕ ǫ , M ǫ , Z ǫ , Y ǫ , ¯ N (cid:17) , ǫ > o is tight in Π. Let (cid:16) ϕ, , Z, Y, ¯ N (cid:17) be any limit point of the tight family. By the Skorohod’s embeddingtheorem, there exist a stochastic basis (Ω , F , P ) and, on this basis, Π-valued randomvariables (cid:16) −→ ϕ ǫ , −→ M ǫ , −→ Z ǫ , −→ Y ǫ , −→ N ǫ (cid:17) , (cid:16) −→ ϕ , , −→ Z , −→ Y , −→ N (cid:17) , such that (cid:16) −→ ϕ ǫ , −→ M ǫ , −→ Z ǫ , −→ Y ǫ , −→ N ǫ (cid:17) (respectively (cid:16) −→ ϕ , , −→ Z , −→ Y , −→ N (cid:17) ) has the same law as (cid:16) ϕ ǫ , M ǫ , Z ǫ , Y ǫ , ¯ N (cid:17) (respectively (cid:16) ϕ, , Z, Y, ¯ N (cid:17) ),and (cid:16) −→ ϕ ǫ , −→ M ǫ , −→ Z ǫ , −→ Y ǫ , −→ N ǫ (cid:17) −→ (cid:16) −→ ϕ , , −→ Z , −→ Y , −→ N (cid:17) in Π , P - a.s.. Set −→ X ǫ = x + −→ M ǫ + −→ Z ǫ + −→ Y ǫ and −→ X = x + −→ Z + −→ Y . From the equation satisfied by n(cid:16) ϕ ǫ , M ǫ , Z ǫ , Y ǫ , ¯ N (cid:17) , ǫ > o , we have that −→ X ǫ satisfies the following SPDE d −→ X ǫt = A ( t, −→ X ǫt ) dt + Z X f ( t, −→ X ǫt , z )( −→ ϕ ǫ ( t, z ) − ν ( dz ) dt + ǫ Z X f ( t, −→ X ǫt − , z ) f −→ N ǫ − −→ ϕ ǫ ǫ ( dz, dt ) , here −→ N ϕǫ is defined as (2.3), that is −→ N ϕǫ ((0 , t ] × U ) = Z (0 ,t ] × U Z (0 , ∞ ) [0 ,ϕ ( s,x )] ( r ) −→ N ǫ ( dsdxdr ) , t ∈ [0 , T ] , U ∈ B ( X ) , f −→ N ϕǫ is the compensated Poisson random measure with respect to −→ N ϕǫ .Using the fact that if f n ∈ D ([0 , T ] , R ) and lim n →∞ f n = 0 with the Skorokhodtopology of D ([0 , T ] , R ), then lim n →∞ sup t ∈ [0 ,T ] | f n ( t ) | = 0. We havelim ǫ → sup t ∈ [0 ,T ] k−→ M ǫ ( t ) k V ∗ = 0 , P -a.s. . Notice that lim ǫ → sup t ∈ [0 ,T ] k−→ Z ǫ ( t ) − −→ Z ( t ) k V ∗ = 0 , P -a.s.and lim ǫ → sup t ∈ [0 ,T ] k−→ Y ǫ ( t ) − −→ Y ( t ) k V ∗ = 0 , P -a.s. , we have lim ǫ → sup t ∈ [0 ,T ] k−→ X ǫ ( t ) − −→ X ( t ) k V ∗ = 0 , P -a.s..Finally, following the proof of Proposition 5.1 and Lemma 5.5, we can obtain −→ X is the unique solution of (5.27) with ϕ replaced by −→ ϕ , and there exists a subsequence ̟ k that lim ̟ k → sup t ∈ [0 ,T ] k−→ X ̟ k ( t ) − −→ X ( t ) k H = 0 , P -a.s.which implies this theorem.We have finished to verify the second part of Condition 2.1. To obtain the first partof Condition 2.1, we just need to replace ǫ R X f ( t, e X ǫt − , z ) e N ǫ − ϕ ǫ ( dz, dt ) by 0 in (4.4)and replacing ϕ ǫ by deterministic elements g n in in the proof of Lemma 4.4–Lemma5.1, then we can similarly prove the following result. Theorem 6.2.
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