Stochastic Navier–Stokes Equations Driven by Lévy Noise in Unbounded 3D Domains

Martingale solutions of the stochastic Navier–Stokes equations in 2D and 3D possibly unbounded domains, driven by the Lévy noise consisting of the compensated time homogeneous Poisson random measure and the Wiener process are considered. Using the classical Faedo–Galerkin approximation and the compactness method we prove existence of a martingale solution. We prove also the compactness and tightness criteria in a certain space contained in some spaces of càdlàg functions, weakly càdlàg functions and some Fréchet spaces. Moreover, we use a version of the Skorokhod Embedding Theorem for nonmetric spaces.


Introduction
Let O ⊂ R d be an open connected possibly unbounded subset with smooth boundary ∂O, where d = 2, 3. We will consider the Navier-Stokes equations in O, with the incompressibility condition the initial condition and with the homogeneous boundary condition u |∂O = 0. In this problem u = u(t, x) = (u 1 (t, x), ...u d (t, x)) and p = p(t, x) represent the velocity and the pressure of the fluid, respectively. Furthermore, f stands for the deterministic external forces. The terms Y F(t, u)η(dt, dy), whereη is a compensated time homogeneous Poisson random measure on a certain measurable space (Y, Y ), and G(t, u(t)) dW(t), where W is a cylindrical Wiener process on some separable Hilbert space Y W , stand for the random forces.
The solutions u n to the Galerkin scheme generate a sequence of laws {L (u n ), n ∈ N} on appropriate functional spaces. To prove that this sequence of probability measures is weakly compact we need appropriate tightness criteria. We concentrate first on the compactness and tightness criteria. If the domain O is unbounded, then the embedding V ⊂ H is not compact. However using Lemma 2.5 in [16], see Appendix C, we can find a separable Hilbert space U such that U ⊂ V, the embedding being dense and compact.
We consider the intersection Using the compactness criterion in the space of càdlàg functions, we prove that a set K is relatively compact in L q if the following three conditions hold Here w [0,T],U (u; δ) stands for the modulus of the function u : [0, T] → U . The above result is a straightforward generalization of the compactness results of [9] and [25]. In the paper [25] the analogous result is proved in the case when the embedding V ⊂ H is dense and compact (in the Banach space setting). In [9] the embedding V ⊂ H is only dense and continuous. However, instead of the spaces of càdlàg functions, appropriate spaces of continuous functions are used. The present paper generalizes both [9] and [25] in the sense that the embedding V ⊂ H is dense and continuous and appropriate spaces of càdlàg functions are considered, i.e. D([0, T]; U ) and D([0, T], H w ). This approach were strongly inspired by the results due to Métivier and Viot, especially the choice of the spaces D([0, T]; U ) and D([0, T], H w ), see [23] and [22]. It is also closely related to the result due to Mikulevicius and Rozovskii [24] and to the classical Dubinsky compactness criterion, [28]. However, both in [28] and [24], the spaces of continuous functions are used.
Using the above deterministic compactness criterion and the Aldous condition in the form given by Joffe and Métivier [19], see also [22], we obtain the corresponding tightness criterion for the laws on the space Z q , see Corollary 1.
We will prove that the set of probability measures induced by the Galerkin solutions is tight on the space Z , where which is not metrizable. Further construction a martingale solutions is based on the Skorokhod Embedding Theorem in nonmetric spaces. In fact, we use the result proved in [25] and following easily from the Jakubowski's version of the Skorokhod Theorem [18] and the version of the Skorokhod Theorem due to Brzeźniak and Hausenblas [6], see Appendix B. This will allow us to construct a stochastic process u with trajectories in the space Z , a time homogeneous Poisson random measurē η and a cylindrical Wiener procesW defined on some filtered probability space (¯ ,F ,P,F) such that the system (¯ ,F ,P,F,η,W,ū) is a martingale solution of the problem (1)(2)(3). In fact,ū is a process with trajectiories in the space Z . In particular, the trajectories ofū are weakly càdlàg ifū is considered as a H-valued process and càdlàg in the bigger space U . The Navier-Stokes equations driven by the compensated Poisson random measure in the 3D bounded domains were studied in Dong and Zhai [15]. The authors consider the martingale problem associated to the Navier-Stokes equations, i.e. a solution is defined to be a probability measure satisfying appropriate conditions, see Definition 3.1 in [15]. The 2D Navier-Stokes equations were considered in [13,14] and [29]. In the present paper, using a different approach we generalize the existence resuls to the case of unbounded 2D and 3D domains. Moreover, we consider more general noise term.
Stochastic Navier-Stokes equations in unbounded 2D and 3D domains were usually considered with the Gaussian noise term, see e.g. [8,11,12] and [9]. Martingale solutions of the stochastic Navier-Stokes equations driven by white noise in the whole space R d , (d ≥ 2), are investigated in [24].
The present paper is organized as follows. In Section 2 we recall basic definitions and properties of the spaces and operators appearing in the Navier-Stokes equations. Section 3 is devoted to the compactness and tightness results. Some auxilliary results about the Aldous condition and tightness are contained in Appendix A. Precise statement of the Navier-Stokes problem driven by Lévy noise is contained in Section 4.2. The main Theorem about existence of a martingale solution of the problem (1-3) is proved in Section 5. Some versions the Skorokhod Embedding Theorems are recalled in Appendix B. In Appendix C we recall Lemma 2.5 in [16] together with the proof.
In the space H we consider the scalar product and the norm inherited from L 2 (O, R d ) and denote them by (·|·) H and | · | H , respectively, i.e.
(u|v) H := (u|v) L 2 , In the space V we consider the scalar product inherited from the Sobolev space where and the norm where u 2 := ∇u 2 L 2 .

The Form b
Let us consider the following three-linear form, see Temam [27], We will recall those fundamental properties of the form b that are valid both in bounded and unbounded domains. By the Sobolev embedding Theorem, see [1], and the Hőlder inequality, we obtain the following estimates for some positive constant c. Thus the form b is continuous on V, see also [27]. Moreover, if we define a bilinear map B by B(u, w) := b (u, w, ·), then by inequality (9) we infer that B(u, w) ∈ V for all u, w ∈ V and that the following inequality holds Moreover, the mapping B : V × V → V is bilinear and continuous. Let us also recall the following properties of the form b , see Temam [27], Lemma II.
In particular, Let us, for any m > 0 define the following standard scale of Hilbert spaces If m > d 2 + 1 then by the Sobolev embedding Theorem, see [1], for some constant c > 0. Thus, b can be uniquely extented to the three-linear form (denoted by the same letter) At the same time the operator B can be uniquely extended to a bounded bilinear operator In particular, it satisfies the following estimate See Vishik and Fursikov [28]. We will also use the following notation, B(u) := B(u, u). Let us also recall the well known result that the map B : V → V is locally Lipschitz continuous, i.e. for every r > 0 there exists a constant L r such that

The Space U and Some Operators
We recall operators and their properties used in [9]. Here we also recall the definition of a Hilbert space U compactly embedded in appropriate space V m . This is possible thanks to the result due to Holly and Wiciak, [16] which we recall with the proof in Appendix C, see Lemma 10. This space will be of crucial importance in further investigations.
Consider the natural embedding j : V → H and its adjoint j * : H → V. Since the range of j is dense in H, the map j * is one-to-one. Let us put and where ((·|·)) is defined by Eq. 7. Let us notice that if u ∈ V, then A u ∈ V and Indeed, this follows immediately from Eq. 8 and the following inequalities (a) For any u ∈ D(A) and v ∈ V: where I stands for the identity operator on H and | is the standard duality pairing. In particular, Proof To prove assertion (a), let u ∈ D(A) and v ∈ V. Then Let us move to the proof of part (b). Since V is dense in H, it is sufficient to prove that D(A) is dense in V. Let w ∈ V be an arbitrary element orthogonal to D(A) with respect to the scalar product in V. Then On the other hand, by (a) and Eq. 6, (u|w) V = (Au|w) H for u ∈ D(A). Hence (Au|w) H = 0 for u ∈ D(A). Since A : D(A) → H is onto, we infer that w = 0, which completes the proof.
Let us assume that m > 1. It is clear that V m is dense in V and the embedding j m : V m → V is continuous. Then by Lemma 10 in Appendix C, there exists a Hilbert space U such that U ⊂ V m , U is dense in V m and the natural embedding ι m : U → V m is compact .
Then we have Since the embedding ι m is compact, ι m is compact as well. Consider the composition and its adjoint Note that ι is compact and since the range of ι is dense in H, ι * : H → U is one-toone. Let us put It is clear that L : D(L) → H is onto. Let us also notice that By equality (19) and the densiness of U in H, we infer similarly as in the proof of assertion (b) in Lemma 1 that D(L) is dense in H. Moreover, for u ∈ D(L), where A is defined by Eq. 14.
Since L is self-adjoint and L −1 is compact, there exists an orthonormal basis {e i } i∈N of H composed of the eigenvectors of operator L. Let us fix n ∈ N and let P n be the operator from U to span{e 1 , ..., e n } defined by where ·|· denotes the duality pairing between the space U and its dual U . Note that the restriction of P n to H, denoted still by P n , is given by and thus it is the (·|·) H -orthogonal projection onto span{e 1 , ..., e n }. Restrictions of P n to other spaces considered in Eq. 17 will also be denoted by P n . Moreover, it is easy to see that It is easy to prove that the system ei ei U n∈N is the (·|·) U -orthonormal basis in the space U and that the restriction of P n to U is the (·|·) U -projection onto the subspace span{e 1 , ..., e n }. In particular, for every u ∈ U See Lemma 2.4 in [9] for details.
We will use the basis {e i } i∈N and the operators P n in the Faedo-Galerkin approximation.  This topology is metrizable by the following metric δ T

The Space of Càdlàg Functions
where T is the set of increasing homeomorphisms of [0, T]. Moreover, D([0, T]; S), δ T is a complete metric space, see [19]. Definition 1 (see [22]) Let u ∈ D([0, T]; S) and let δ > 0 be given. A modulus of u is defined by where δ is the set of all increasing sequencesω = {0 = t 0 < t 1 < ... < t n = T} with the following property If no confusion seems likely, we will denote the modulus by w [0,T] (u, δ).
We have the following criterion for relative compactness of a subset of the space D([0, T]; S), see [19,22], Ch.II, and [4], Ch.3, analogous to the Arzelà-Ascoli Theorem for the space of continuous functions.

Deterministic Compactness Criterion
Let us recall that V and H are Hilbert spaces defined by Eqs. 4-8. Since O is an arbitrary domain of R d , (d = 2, 3), the embedding V → H is dense and continuous. We have defined a Hilbert space U ⊂ V such that the embedding U → V is dense and compact, see Eq. 16. In particular, we have with appropriate scalar products and norms, i.e.
and |u| 2 The symbols H OR and V OR will stand for the corresponding dual spaces.
Since the sets O R are bounded, Let q ∈ (1, ∞). Let us consider the following three functional spaces, analogous to those considered in [25] and [9], see also [22,23]: with the topology T 3 generated by the seminorms Let H w denote the Hilbert space H endowed with the weak topology. Let us consider the fourth space, see [25], Let B w denote the ball B endowed with the weak topology. It is well-known that the B w is metrizable, see [5]. Let q r denote the metric compatible with the weak topology on B. Let us consider the following space  We recall the proof in Appendix E.
The following Theorem is a generalization of the results of [9] and [25]. In the paper [25] the analogous result is proved in the case when the embedding V ⊂ H is dense and compact. In [9] the embedding V ⊂ H is only dense and continuous. However, instead of the spaces of càdlàg functions, appropriate spaces of continuous functions are used. The following result generalizes both [9] and [25] in the sense that the embedding V ⊂ H is dense and continuous and appropriate spaces of càdlàg functions are considered, i.
Theorem 2 Let q ∈ (1, ∞) and let and let T be the supremum of the corresponding topologies. Then a set K ⊂ Z q is T -relatively compact if the following three conditions hold Proof We can assume that K is a closed subset of Z q . Because of the assumption (b), the weak topology in L q w (0, T; V) induced on Z q is metrizable. Since the topology in L q (0, T; H loc ) is defined by the countable family of seminorms (24), this space is also metrizable. By assumption (a), it is sufficient to consider the metric subspace D([0, T]; B w ) ⊂ D([0, T], H w ) defined by Eqs. 26 and 27 with r := sup u∈K sup s∈[0,T] |u(s)| H . Thus compactness of a subset of Z q is equivalent to its sequential compactness. Let (u n ) be a sequence in K . By the Banach-Alaoglu Theorem condition (b) yields that the set K is compact in L q w (0, T; V). Using the compactness criterion in the space of càdlàg functions contained in Theorem 1, we will prove that ( Therefore there exists a subsequence (u nk ) ⊂ (u n ) such that in U for all continuity points of function u, (see [4]). By condition (a) and the Lebesgue dominated convergence theorem, we infer that for all p ∈ [1, ∞) We claim that In order to prove it let us fix R > 0. Since, by Eq. 23 the embedding V OR → H OR is compact and the embeddings H OR → H → U are continuous, by the Lions Lemma, [20], for every ε > 0 there exists a costant C = C ε,R > 0 such that Thus for almost all s ∈ [0, T] and so for all k ∈ N Passing to the upper limit as k → ∞ in the above inequality and using the estimate By the arbitrariness of ε, The proof of Theorem is thus complete.

Tightness Criterion
Let us recall that U, V, H are separable Hilbert spaces such that where the embedding U → V is compact and V → H is continuous. Using the compactness criterion formulated in Theorem 2 we obtain the corresponding tightness criterion in the space Z q . Let us first recall that the space Z q is defined by and it is equipped with the topology T , see Eq. 28.

satisf ies the Aldous condition [A] in U .
LetP n be the law of X n on Z q . Then for every ε > 0 there exists a compact subset K ε of Z q such thatP We recall the Aldous condition [A] in Appendix A, see Definition 5. The proof of Corrollary 1 is postponed to Appendix A, as well.

Time Homogeneous Poisson Random Measure
We follow the approach due to Brzeźniak and Hausenblas [6,7], see also [17] and [26] Let ( , F , P) be a complete probability space with filtration F := (F t ) t≥0 satisfying the usual hypotheses, see [21].
is F-adapted and its increments are independent of the past, i.e. if t > s ≥ 0, If η is a time homogeneous Poisson random measure then the formula is an integrable martingale on ( , F , F, P). The random measure l ⊗ ν on B(R + ) ⊗ Y , where l stands for the Lebesgue measure, is called an compensator of η and the difference between a time homogeneous Poisson random measure η and its compensator, i.e.η is called a compensated time homogeneous Poisson random measure.
Let us also recall basic properties of the stochastic integral with respect toη, see [7,17] and [26] for details. Let H be a separable Hilbert space and let P be a

Statement of the Problem
Problem (1-3) can be written as the following stochastic evolution equation Assumption We assume that and for each p ∈ {2, 4, 4 where γ > 0 is some positive constant.
Moreover there exist λ, κ ∈ R and a ∈ 2 − 2 3+γ , 2 such that for some C > 0. Moreover, for every v ∈ V the mappingG v defined by is a continuous mapping from Let us recall that the space L 2 (0, T; H loc ) is defined by Eq. 24. For any Hilbert space •η is a time homogeneous Poisson random measure on (Y, Y ) over ¯ ,F ,F,P with the intensity measure ν, •W is a cylindrical Wiener process on the space Y W over ¯ ,F ,F,P , such that for all t ∈ [0, T] and all v ∈ V the following identity holdsP-a.s.
We will prove existence of a martingale solution of the Eq. 30. To this end we use the Faedo-Galerkin method. The Galerkin approximations generate a sequence of probability measures on appropriate functional space. We will prove that this sequence is tight. Let us emphasize that to prove the tightness, assumption (F.2) with p = 2 in inequality (32) is sufficient. The stronger condition on p, i.e. inequality (32) for a certain p > 4, is connected with the construction of the processū to deal with the nonlinear term. Assumptions (G.2)-(G.3) allow to consider the Gaussian noise term G dependent both on u and ∇u. This corresponds to inequality (35) with a < 2. The case when a = 2 is related to the noise term G dependent on u but not on its gradient. Moreover, assumptions (F.3) and (G.3) are important in the case of unbounded domain O. In the case when O is bounded, they can be omitted, see [25].

Faedo-Galerkin Approximation
be the orthonormal basis in H composed of eigenvectors of the operator L defined by Eq. 18. Let H n := span{e 1 , ..., e n } be the subspace with the norm inherited from H and let P n : H → H n be defined by Eq. 20. Let us fix m > d 2 + 1 and let U be the space defined by Eq. 16. Consider the following mapping Since H n ⊂ H, B n is well defined. Moreover, B n : H n → H n is globally Lipschitz continuous.
Let us consider the classical Faedo-Galerkin approximation in the space H n Lemma 3 For each n ∈ N, there exists a unique F-adapted, càdlàg H n valued process u n satisfying the Galerkin (38).
Proof The assertion follows from Theorem 9.1 in [17].
Using the Itô formula, see [17] or [21], and the Burkholder-Davis-Gundy inequality, see [26], we will prove the following lemma about a priori estimates of the solutions u n of Eq. 38. In fact, these estimates hold provided the noise terms satisfy only condition (32) in assumption (F.2) and condition (35) in assumption (G.2).

Lemma 4
The processes (u n ) n∈N satisfy the following estimates.
(ii) There exists a positive constant C 2 such that Let us recall that γ > 0 is defined in assumption (F.2).
Proof For all n ∈ N and all R > 0 let us define Since the process u n (t) t∈[0,T] is F-adapted and right-continuous, τ n (R) is a stopping time. Moreover, since the process (u n ) is càdlàg on [0, T], the trajectories t → u n (t) are bounded on [0, T], P-a.s. Thus τ n (R) ↑ T, P-a.s., as R ↑ ∞.
Assume first that p = 2 or p = 4 + γ . Using the Itô formula to the function φ(x) := |x| p := |x| By Eqs. 11 and 15 we obtain for all t ∈ [0, T] Let us recall that according to Eq. 15 we have A u|u = ((u|u)) and thus Hence inequality (35) in assumption (G.2) can be written equivalently in the following form Tr P n G(s, u n (s)) ∂ 2 φ ∂ x 2 P n G(s, u n (s)) * ds Moreover, by assumption (A.1), Eq. 8 and the Schwarz inequality, we obtain for every ε > 0 and for all s and hence by the Young inequality 2 for some constants c, c 1 > 0. Thus Let us choose ε > 0 such that p − pε − 1 2 p( p − 1)(2 − a) > 0, or equivalently, Note that since by assumption (G.2) a ∈ 2 − 2 3+γ , 2], such an ε exists.
By Eqs. 43 and 32 we obtain the following inequalities for some constantc p > 0. Thus by the Fubini Theorem, we obtain the following inequality In particular, for some positive constant C p . Passing to the limit as R ↑ ∞ and using again the Fatou Lemma we infer that In particular, putting p := 2 by Eqs. 8, 49 and 47 we obtain assertion (40). Let us move to the proof of inequality (39). By the Burkholder-Davis-Gundy inequality we obtain for some positive constants C i , i = 1, ..., 4. By Eq. 51 and the Young inequality we infer that for some positive constants K 1 and K 2 . Thus |M n (r ∧ τ n (R))| p |u n (s)| p−2 u n (s)P n G(s, u n (s)) dW(s) By inequality (35) in assumption (G.2) and estimates (49), (47) we have the following inequalities where I n is defined by Eq. 44. Since inequality (55) holds for all t ∈ [0, T] and the right-hand side of Eq. 55 is independent of t, we infer that Using inequalities (47), (53), (45) and (54) in Eq. 56 we infer that for some constant C 1 ( p) independent of n ∈ N and R > 0. Passing to the limit as R → ∞, we obtain inequality (39 Since n ∈ N was chosen in an arbitray way, we infer that . The proof of Lemma is thus complete.

Tightness
Let m > d 2 + 1 be fixed and let U be the space defined by Eq. 16. We will apply Corollary 1 with q := 2. So, let us consider the space For each n ∈ N, the solution u n of the Galerkin equation defines a measure L (u n ) on (Z , T ). Using Corollary 1 we will prove that the set of measures L (u n ), n ∈ N is tight on (Z , T ). The inequalities (39) and (40) in Lemma 4 are of crucial importance. However, to prove tightness it is sufficient to use inequality (39) only with p = 2.

Lemma 5
The set of measures L (u n ), n ∈ N is tight on (Z , T ).
Proof We will apply Corollary 1. By estimates (39) and (40), conditions (a), (b) are satisfied. Thus, it is sufficient to prove that the sequence (u n ) n∈N satisfies the Aldous condition [A] in the space U . We will use Lemma 9 in Appendix A. Let (τ n ) n∈N be a sequence of stopping times such that 0 ≤ τ n ≤ T. By Eq. 38, we have Let θ > 0. We will check that each term J n i , i=1,...,6, satisfies condition (89) in Lemma 9.
Since A : V → V and |A (u)| V ≤ u and the embedding V → U is continuous, by the Hőlder inequality and Eq. 40, we have the following estimates Thus J n 2 satifies condition (89) with α = 1 and β = 1 2 . Let us consider the term J n 3 . Since m > d 2 + 1 and U → V m , by Eqs. 12 and 39 we have the following inequalities where B stands for the norm of B : H × H → V m . This means that J n 3 satisfies condition (89) with α = β = 1.
Let us move to the term J n 4 . By the Hőlder inequality, we have Hence condition (89) holds with α = 1 and β = 1 2 . Let us consider the term J n 5 . Since H → U , by Eq. 29, condition (32) with p = 2 in Assumption (F.2) and by Eq. 39, we obtain the following inequalities Thus J n 5 satisfies condition (89) with α = 2 and β = 1. Let us consider the term J n 6 . By the Itô isometry, condition (36) in assumption (G.3), continuity of the embedding V → U and inequality (39), we have Thus J n 6 satisfies condition (89) with α = 2 and β = 1. By Lemma 9 the sequence (u n ) n∈N satisfies the Aldous condition in the space U . This completes the proof of Lemma.
We will now move to the proof of the main Theorem of existence of a martingale solution. The main difficulties occur in the term containing the nonlinearity B and in the noise terms F and G. To deal with the nonlinear term, we need inequality (39) for some p > 4. Moreover, we will see that the sequence (ū n ) of approximate solutions is convergent in the Fréchet space L 2 (0, T; H loc ). So, we will use the property of the mapping B contained in Lemma 6 below. Analogous problems appear in the noise terms, where assumptions (F.3) and (G.3) will be needed in the case when the domain O is unbounded. For simplicity we assume that dim Y W = 1, i.e. we consider onedimensional cylindrical Wiener process W(t), t ∈ [0, T]. Construction of a martingale solution is based on the Skorokhod Theorem for nonmetric spaces. The method is closely related to the approach due to Brzeźniak and Hausenblas [6].
We will denote this sequences again by ((u n , η n , W n )) n∈N and (ū n ,η n ,W n ) n∈N . Moreover,η n , n ∈ N, and η * are time homogeneous Poisson random measures on (Y, Y ) with intensity measure ν andW n , n ∈ N, and W * are cylindrical Wiener processes, see [6,Section 9]. Using the definition of the space Z , see Eq. 57, in particular, we havē Since the random variablesū n and u n are identically distributed, we have the following inequalities. For every p ∈ [1, and Let us fix v ∈ U. Analogously to [6], let us denote and Step 1 0 We will prove that To prove Eq. 63 let us write Since by Eq. 58ū n → u * in D([0, T]; U ) and by Eq. 59 sup t∈[0,T] |ū n (t)| 2 H < ∞,P-a.s. and the embedding H → U is continuous, by the Dominated Convergence Theorem we infer thatP-a.s.,ū n → u * in L 2 (0, T; U ). Then Moreover, by the Hőlder inequality and Eq. 59 for every n ∈ N and every r ∈ 1, 2 + γ for some constants c,c > 0. By Eqs. 66, 65 and the Vitali Theorem we infer that i.e. Eq. 63 holds. Let us move to the proof of Eq. 64. Note that by the Fubini Theorem, we have We will prove that each term on the right hand side of Eq. 61 tends in L 2 ([0, T] ×¯ ) to the corresponding term in Eq. 62. Since by Eq. 58ū n → u * in D(0, T; H w )P-a.s. and u * is continuous at t = 0, we infer that (ū n (0)|v) H → (u * (0)|v) HP -a.s. By Eq. 59 and the Vitali Theorem, we have Let us move to the nonlinear term. We will use the following auxiliary result proven in [9]. ( It is easy to see that for sufficiently large n ∈ N B n (ū n (s)) = P n B(ū n (s)), Let us move to the noise terms. Let us assume first that whereF v is the mapping defined by Eq. 33. Since by Eq. 58ū n → u * in L 2 (0, T; H loc ), P-a.s., by assumption (F. Moreover, by inequality (32) in assumption (F.2) and by Eq. 59 for every t ∈ [0, T] every r ∈ 1, 2 + γ 2 and every n ∈ N the following inequalities hold Let now v ∈ H and let ε > 0. Since V is dense in H, there exists v ε ∈ V such that |v − v ε | 2 H < ε. By Eq. 32 the following inequalities hold t 0 Y F(s,ū n (s − ); y) − F(s, u * (s − ); y)|v 2 dν(y)ds Hence by Eq. 59 Since ε > 0 was chosen in an arbitrary way, we infer that for all v ∈ H Moreover, since the restriction of P n to the space H is the (·|·) H -projection onto H n , see Section 2.3, we infer that also Hence by the properties of the integral with respect to the compensated Poisson random measure and the fact thatη n = η * , we have Moreover, by inequality (32) in assumption (F.2) and by Eq. 59 we obtain the following inequalities Since U ⊂ H, Eq. 80 holds for all v ∈ U, as well.
Let us move to the second part of the noise. Let us assume first that v ∈ V . We have t 0 G(s,ū n (s)) − G(s, u * (s))|v 2 LHS(Yw;R) ds whereG v is the mapping defined by Eq. 37. Since by Eq. 58ū n → u * in L 2 (0,T;H loc ), P-a.s., by the second part of assumption (G. Moreover, by Eqs. 36 and 59 we see that for every t ∈ [0, T] every r ∈ 1, 2 + γ 2 and every n ∈ N E t 0 G(s,ū n (s)) − G(s, u * (s))|v 2 LHS(Yw;R) ds for some positive constants c, c 1 ,c. Thus by Eqs. 81, 82 and the Vitali Theorem for some c > 0. Thus by Eq. 59 we obtain the following inequalities Thus by inequality (36) in assumption (G.3) and by Eq. 59 we obtain for some c > 0. By Eqs. 85, 86 and the Dominated Convergence Theorem we infer that By Eqs. 67, 70, 74, 80 and 87 the proof of Eq. 64 is complete.
Step 2 0 Since u n is a solution of the Galerkin equation, for all t ∈ [0, T] In particular, Since L (u n , η n , W n ) = L (ū n ,η n ,W n ), Moreover, by Eqs. 63 and 64 Hence for l-almost all t ∈ [0, T] andP-almost all ω ∈¯ Since u * is Z -valued random variable, in particular u * ∈ D([0, T]; H w ), i.e. u * is weakly càdlàg. Hence the function on the left-hand side of the above equality is càdlàg with respect to t. Since two càdlàg functions equal for l-almost all t ∈ [0, T] must be equal for all t ∈ [0, T], we infer that for all t ∈ [0, T] and all v ∈ U Since U is dense in V, we infer that the above equality holds for all v ∈ V. Puttinḡ u := u * ,η := η * andW := W * , we infer that the system (¯ ,F ,P,F,ū,η,W) is a martingale solution of the Eq. (30). The proof of Theorem 3 is thus complete.
Definition 4 (See [19]) We say that the sequence (X n ) of S-valued random variables satifies condition [T] iff Let us recall that w [0,T] stands for the modulus defined by Eq. 21.

Remark
Proof Fix ε > 0. By [T], for each k ∈ N there exists δ k > 0 such that k and let A ε := ∞ k=1 B k . We assert that for each n ∈ N P n (A ε ) ≥ 1 − ε.
Indeed, we have the following estimate To prove Eq. 88, let us fixε > 0. Directly from the definition of A ε , we infer that sup u∈Aε w [0,T] (u, δ k ) ≤ 1 k for each k ∈ N. Choose k 0 ∈ N such that 1 k0 ≤ε and let δ 0 := δ k0 . Then for every δ ≤ δ 0 we obtain which completes the proof of Eq. 88 and the proof of Lemma. Now, we recall the Aldous condition which is connected with condition [T] (see [19,22] and [2]). This condition allows to investigate the modulus for the sequence of stochastic processes by means of stopped processes.
In the following Remark we formulate a certain condition which guaranties that the sequence (X n ) n∈N satisfies condition [A].

Lemma 9
Let (E, · E ) be a separable Banach space and let (X n ) n∈N be a sequence of E-valued random variables. Assume that for every sequence (τ n ) n∈N of F-stopping times with τ n ≤ T and for every n ∈ N and θ ≥ 0 the following condition holds for some α, β > 0 and some constant C > 0. Then the sequence (X n ) n∈N satisf ies condition [A] in the space E.
Proof Let us fix ε > 0 and η > 0. By the Chebyshev inequality for every n ∈ N and every θ > 0 we have Let δ := η α ε C 1 β . Let us fix n ∈ N. Then for every θ ∈ [0, δ] we have the following Hence sup 0≤θ ≤δ Since the above inequality holds for every n ∈ N, one has By the Chebyshev inequality and by (b), we infer that for any r > 0 Let R 2 be such that C2 Then Let B 2 := u ∈ Z q : u L q (0,T;V) ≤ R 2 . By Lemmas 7 and 8 there exists a subset A ε It is sufficient to define K ε as the closure of the set B 1 ∩ B 2 ∩ A ε 3 in Z q . By Theorem 2, K ε is compact in Z q . The proof is thus complete.

Appendix B: The Skorokhod Embedding Theorems
Let us recall the following Jakubowski's version of the Skorokhod Theorem [18], see also Brzeźniak and Ondreját [10].
Theorem 4 (Theorem 2 in [18]) Let (X , τ ) be a topological space such that there exists a sequence ( f m ) of continuous functions f m : X → R that separates points of X . Let (X n ) be a sequence of X valued random variables. Suppose that for every ε > 0 there exists a compact subset K ε ⊂ X such that Then there exists a subsequence (X nk ) k∈N , a sequence (Y k ) k∈N of X valued random variables and an X valued random variable Y def ined on some probability space ( , F , P) such that and forall ω ∈ : We will use the following version of the Skorokhod Theorem due to Brzeźniak and Hausenblas [6].
Let ( , F , P) be a probability space and let χ n : → E 1 × E 2 , n ∈ N, be a family of random variables such that the sequence {L aw(χ n ), n ∈ N} is weakly convergent on E 1 × E 2 . Finally let us assume that there exists a random variable ρ : → E 1 such that L aw(π 1 • χ n ) = L aw(ρ), ∀ n ∈ N.
Then there exists a probability space (¯ ,F ,P), a family of E 1 × E 2 -valued random variables {χ n , n ∈ N} on (¯ ,F ,P) and a random variable χ * :¯ → E 1 × E 2 such that Remark Theorem 5 remains true if we substitute the Banach spaces E 1 , E 2 by the separable complete metric spaces.
Using the ideas due to Jakubowski [18], we can proof the following generalization of Theorem 5 to the case of nonmetric spaces. Let us notice that in comparison to Theorem 5 we will assume that the sequence {L aw(χ n ), n ∈ N} is tight. The assumption of the weak convergence of {L aw(χ n ), n ∈ N} is not sufficient in the case of nonmetric spaces, see [18]. Corollary 2 (Corollary 5.3 in [25]) Let X 1 be a separable complete metric space and let X 2 be a topological space such that there exists a sequence { f ι } ι∈N of continuous functions f ι : X 2 → R separating points of X 2 . Let X := X 1 × X 2 with the Tykhonof f topology induced by the projections Let ( , F , P) be a probability space and let χ n : → X 1 × X 2 , n ∈ N, be a family of random variables such that the sequence {L aw(χ n ), n ∈ N} is tight on X 1 × X 2 . Finally let us assume that there exists a random variable ρ : → X 1 such that L aw(π 1 • χ n ) = L aw(ρ) for all n ∈ N.
Then there exists a subsequence χ nk k∈N , a probability space (¯ ,F ,P), a family of X 1 × X 2 -valued random variables {χ k , k ∈ N} on (¯ ,F ,P) and a random variable For the convenience of the reader we recall the proof.
Proof Using the ideas due to Jakubowski [18], the proof can be reduced to Theorem 5. Let us denote where χ i n := π i • χ n , i = 1, 2. Since the sequence {L aw(χ n ), n ∈ N} is tight on X 1 × X 2 , we infer that the sequence {L aw(χ 2 n ), n ∈ N} is tight on X 2 . Let K m ⊂ X 2 be compact subsets such that K m ⊂ K m+1 , m = 1, 2, ... and Let us consider the mappingf : On the set R N let us define the function Furthermore, the sequence of laws L aw(f • χ 2 n , •f • χ 2 n ), n ∈ N is tight on R N × N.
Let us consider the product space X 1 × (R N × N) and let P 1 := X 1 × (R N × N) → X 1 be the projection onto X 1 and P 2 := X 1 × (R N × N) → R N × N be the projection onto R N × N. Moreover let ξ n , n ∈ N, be X 1 × (R N × N)-valued random variables defined by where ξ 1 n := χ 1 n and Remark that the sequence of laws L aw(ξ n ), n ∈ N is tight on X 1 × (R N × N).
By the Prokhorov Theorem we can choose a subsequence (n k ) k∈N such that L aw(ξ nk ), k ∈ N is weakly convergent on X 1 × (R N × N). Thus the subsequence ξ nk k∈N satisfies the assumption of Theorem 5. Hence there exists a probability space (¯ ,F ,P), a family of X 1 × (R N × N)-valued random variables {ξ k , k ∈ N} on (¯ ,F ,P) and a random variable ξ * :¯ → X 1 × (R N × N) such that (i) L aw(ξ k ) = L aw(ξ nk ) for all k ∈ N; The linear operator is well defined. Moreover, l is an injection and hence we may introduce the following inner product Now, l is an isometry onto the pre-Hilbert space (H , (·| · ) H ) and consequently H is (·| · ) H -complete. Let us notice that for all x, y ∈ H Thus, for any k, n ∈ N, k < n, we have the following estimate Since in particular, the sequence s n := n i=1 (x|h i )h i is Cauchy in the Banach space ( , | · | ), there exists ϕ ∈ such that lim n |s n − ϕ| = 0. On the other hand, s n = n i=1 (x|h i )h i → x in H. Thus by the uniqueness of the limit ϕ = x ∈ and Thus H ⊂ continuously (with the norm of the embedding not exceeding 1 − η 0 ). We will show that the embedding j : H → is compact. It is sufficient to prove that the ball Z := {x ∈ H : |x| H ≤ 1} is relatively compact in ( , | · | ). According to the Hausdorff Theorem it is sufficient to find (for every fixed ε) an ε-net of the set j(Z ).
Since lim n→∞ η n = 1, there exists n ∈ N such that 1 − η n ≤ ε 2 . The linear operator being finite-dimensional is compact. Therefore S n (Z ) is relatively compact in ( , | · | ) and consequently there is a finite subset F ⊂ such that S n (Z ) ⊂ ϕ∈Z B (ϕ, ε 2 ). We will show that the set F is the ε-net for j(Z ). Indeed, let x ∈ Z . Then S N (x) → x in ( , | · | ) and On the other hand, S n (x) ∈ S n (Z ), so, there is ϕ ∈ F such that S n (x) ∈ B (ϕ, ε 2 ). Finally, i.e. x ∈ B (ϕ, ε). Thus The proof is thus complete.

Appendix D: Proof of Lemma 6
Proof Assume first that ψ ∈ V . Then there exists R > 0 such that suppψ is a compact subset of O R . Then, using the integration by parts formula, we infer that for every v, w ∈ H We have B(u n , u n ) − B(u, u) = B(u n − u, u n ) + B(u, u n − u). Thus, using the estimate (92) and the Hőlder inequality, we obtain If ψ ∈ V m then for every ε > 0 there exists ψ ε ∈ V such that ψ − ψ ε Vm ≤ ε. Then Let (t k ) k∈N ⊂ [0, T] be a sequence convergent to t − 0 . Since |u n (t k )| H ≤ r, by the Banach-Alaoglu Theorem there exists a subsequence convergent weakly in H to some b ∈ H, i.e. there exists (t kl ) l∈N such that u n (t kl ) → b weakly in H as l → ∞. Since the embedding H → U is continuous, we infer that u n (t kl ) → b weakly in U as l → ∞.
On the other hand, by Eq. 94 u n (t kl ) → a in U as l → ∞.
By Eqs. 94 and 95 we infer that lim t→t0 (u n (t) − a|h) H = 0. Now, let h ∈ H and let ε > 0. Since U is dense in H, there exists h ε ∈ U such that |h − h ε | H ≤ ε. We have the following inequalities Passing to the upper limit as t → t − 0 , we obtain lim sup Since ε was chosen in an arbitrary way, we infer that