Stochastic Reaction-diffusion Equations Driven by Jump Processes

We establish the existence of weak martingale solutions to a class of second order parabolic stochastic partial differential equations. The equations are driven by multiplicative jump type noise, with a non-Lipschitz multiplicative functional. The drift in the equations contains a dissipative nonlinearity of polynomial growth.


Introduction
The aim of this paper is to study a class of stochastic reaction-diffusion equations driven by a Lévy noise. A motivating example is the following stochastic partial differential equation with a Dirichlet boundary condition ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ du(t, ξ ) = u(t, ξ ) dt + u(t, ξ ) − u(t, ξ ) 3 dt where L = {L(t) : t ≥ 0} is a real valued Lévy process whose Lévy measure ν has finite p-moment for some p ∈ (1,2] and O ⊂ R d is a bounded domain with smooth boundary. One consequence of our main results is that for every u 0 ∈ C 0 (O) there exists a C 0 (O)valued process u = {u(t) : t ≥ 0}, which is a martingale solution to problem (1.1). The diffusion coefficient in Eq. 1.1, i.e., the function g(u) = √ |u|/(1 + √ |u|), is just a simple example of a bounded and continuous function, the drift term f (u) = u − u 3 is an example of a dissipative function f : R → R of polynomial growth, and the Laplace operator is a special case of a second order operator. Our results allow us to treat equations with more general coefficients than those in the model problem (1.1). In fact we can study stochastic partial differential equations with second order uniformly elliptic dissipative operators with non-constant coefficients and an unbounded nonlinear map as drift term. Moreover, our results are applicable to equations with infinite-dimensional Lévy processes as well as systems with more general initial data, for instance, elements of the Lebesgue or Sobolev spaces L q (O) or W γ,q 0 (O). The details are presented in Sections 4, 5 and 6. In our paper we adopt the approach used in [10] by the first named author and Gatarek, in which a similar problem but driven by a Wiener process was treated. The major differences of the current work with respect to [10] are as follows. Firstly, the approach [10] heavily depends on the use of the theory of Itô integral in martingale type 2 Banach spaces with respect to a cylindrical Wiener process, while we use an approach which relies on stochastic integration in martingale type p Banach spaces with respect to a Poisson random measure, see [13]. Secondly, the compactness argument in [10] relies on the Hölder continuity of the trajectories of the corresponding stochastic convolution process. However, the trajectories of the stochastic convolution process driven by a Lévy process are not continuous, then the use of the counterpart of the Hölder continuity, i.e. the càdlàg property of the trajectories seems natural to be used. Unfortunately, as many counterexamples have shown, see for instance a recent monograph [65] as well as the recent papers [19] and [11], the trajectories of the stochastic convolution processes driven by a Lévy process may not even be càdlàg in the space in which the Lévy process lives. Hence, this issue has to be handled with special care. Thirdly, we do not use martingale representation theorem, unlike the authors in [10] who used a result by Dettweiler [27]. Instead we use an ad hoc method based on a generalisation of the Skorohod representation theorem, see Theorem C.1. Finally, in order to control certain norms of the approximate solutions, instead of using stopping times as in [10], we apply interpolation methods. Summarizing, this paper contains general results on the existence of martingale solutions to stochastic reaction-diffusion equations with dissipative type coefficients of polynomial growth and with multiplicative Lévy noise. We should also point out that, as a by-product of our results, we are able to fill a gap in the proof of the main result in the article [43]. The gap is related to the Step I, see page 8, in the current paper.
Our paper confirms an observation that has already been made in earlier papers [13,42] that the theory of stochastic integration with respect to a Poisson random measure in martingale type p Banach spaces is, to a large extend, analogous to the theory of stochastic integration with respect to a cylindrical Wiener process in martingale type 2 Banach spaces provided the space of γ -radonifying operators is replaced by the space L p (ν), where ν is the intensity measure of the Poisson random measure in question. We would also like to point out that the theory of Banach space valued stochastic integrals with respect to a Poisson random measure is richer than the corresponding Gaussian theory, see the recent paper [30] by S. Dirksen. It would be of great interest to develop a theory of inequalities for stochastic convolutions and generalise the paper [13] in the framework of [30].
It is worth mentioning that although SPDEs with unbounded nonlinearity driven by a Lévy process have not been as extensively studied as their Gaussian counterparts, there exists a number of interesting recent publications on the subject. For instance, Truman and Wu [74,75], Jacob et al. [61], and Giri and Hausenblas [41] studied equations with Burgers type nonlinearities driven by a Lévy noise. In addition, Dong and Xu in [31] considered the Burgers equation with the compound Poisson noise, thus in fact dealing with a deterministic Burgers' equation on random intervals and random jumps. Some discussion of stochastic Burgers' equations with additive Lévy noise is contained in [19], where it was shown how integrability properties of trajectories of the corresponding Ornstein-Uhlenbeck process play an important role in the existence and uniqueness of solutions. In the recent paper [32] Dong and Xie studied the stochastic Navier-Stokes equations (NSEs) driven by a Poisson random measure with finite intensity measure. Fernando and Sritharan [37] and the first two named authours together with Zhu [17] studied the 2-D stochastic NSEs by means of a local-monotonicity method of Barbu [4]. This method seems to be restricted to the 2-D NSEs and does not require the use of compactness results. In their beautiful monograph [65] Peszat and Zabczyk studied classes of reaction-diffusion equations driven by an additive Lévy process. In this case, the stochastic evolution equations can be transformed into an evolution equation with random coefficients, a method which usually does not work with multiplicative noise. In another recent paper [55] Marinelli and Röckner investigated a certain class of generalized solutions to problems similar to ours. Röckner and Zhang in [67] established the existence and uniqueness of solution to and a large deviation principles for a class of stochastic evolution equations driven by jump processes. Finally, Debussche et al. in [25,26] considered a stochastic Chafee-Infante equation driven by an additive Lévy noise and investigated the dynamics of the equation, for instance, the first exit times from domains of attraction of the stationary solutions of the deterministic equation.
The stochastic PDEs driven by Lévy processes in Banach spaces have not been intensively studied, apart from a few papers by the second named author, like [42,43], a very recent paper [13] by the first two named authors and [54] by Mandrekar and Rüdiger (who actually studied ordinary stochastic differential equations in martingale type 2 Banach spaces). Martingale solution to SPDEs driven by Lévy processes in Hilbert spaces are not often treated in the literature. Mytnik [60] constructed a weak solution to SPDEs with non-Lipschitz coefficients driven by space-time stable Lévy noise. In [58] Mueller studied non-Lipschitz SPDEs driven by nonnegative stable Lévy noise of index α ∈ (0, 1). Mueller, Mytnik and Stan [59] investigated the heat equation with one-sided time independent stable Lévy noise. One should add that the noise in the paper [58] does not satisfy the hypothesis of the current work.
The current paper is organized as follows. In Section 2 we introduce the notations used later on in the paper and we present the standing hypotheses and essential preliminary facts. Our two main results, i.e Theorems 3.2 and 3.4 are stated in Section 3. In that section we also present Theorem 3.5 which could be seen as the reformulation of some of our main results in terms of a Lévy process and not a Poisson random measure. Three examples illustrating the applicability of our results are presented in Sections 4, 5 and 6. To be more precise, SPDEs with dissipative polynomial drift driven by a real-valued α-stable tempered Lévy process are treated in Section 4, SPDEs with a bounded drift driven by a space-time Lévy white noise are treated in Section 5 and finally, stochastic reaction-diffusion equations with dissipative polynomial growth drift, driven by a space-time Lévy white noise, are treated in Section 6. In Section 7 we state and prove several preliminary results about the stochastic convolution processes which we believe are interesting in themselves. Sections 8 and 9 are devoted to the proofs of our results. Unfortunately, these proofs are very technical and long and hence their brief outline is presented just after the statement of Theorem 3.5, see page 22. In the appendices we recall some definitions and well-known results in analysis and probability theory. We also prove new results, amongst them a modified version of the Skorohod representation theorem, see Theorem C.1, which are interesting in themselves.
We finish the Introduction by pointing out that the approach presented in this paper (or rather it's earlier arXiv version) has already been taken up and used for the proof of the existence of solutions to Stochastic Navier-Stokes equations and second grade fluids driven by Lévy noise, see [57] and [44], respectively. Notation 1 By N we denote the set of natural numbers, i.e., N = {0, 1, 2, · · · } and by N, respectively N * , we denote the set N ∪ {+∞}, respectively N \ {0}. Whenever we speak about N, respectively N-valued measurable functions, we implicitly assume that the set N, respectively N, is equipped with the full σ -field 2 N , resp. 2 N . By R + we denote the set [0, ∞) of nonnegative real numbers and by R * the set R \ {0}. If X is a topological space, then by B(X) we denote the Borel σ -field on X. By Leb we denote the Lebesgue measure on (R d , B(R d )) or (R, B(R)). The space of bounded linear operators from a Banach space Y 1 to a Banach space Y 2 is denoted by L(Y 1 , Y 2 ). The norm of Finally, by Z ⊗ B(R + ) we denote the product σ -field on Z × R + and by ν ⊗ Leb we denote the product measure of ν and the Lebesgue measure Leb.
For a Banach space Y by D([0, T ], Y ) we denote the space of all càdlàg functions u : [0, T ] → Y which we equip with the J 1 -Skorohod topology, i.e., the finest among all Skorohod topologies.

Hypotheses, Notations and Preliminaries
In this section we introduce the notation and hypotheses used throughout the whole paper. Moreover, we present some preliminary results that will be frequently used later on.

Analytic Assumptions and Hypotheses
Let us begin with a list of assumptions. Whenever we use any of them this will be clearly indicated.
Let E be a Banach space and let A be a closed linear densely defined map in E. The norm in the space E is denoted by |·| and the norm of any other Banach space Y is denoted by |·| Y .
In what follows we will frequently use the following assumptions about the Banach space E and the linear map A. Assumption 1 1(i) Assume that E is a separable UMD and type p, for a certain p ∈ (1, 2], Banach space. 1 1(ii) A is a positive operator in E, i.e, a densely defined and closed operator for which there exists M > 0 such that for λ > 0 1(iii) −A is the infinitesimal generator of an analytic semigroup denoted by (e −tA ) t≥0 on E. We also assume that A has compact resolvent. 1(iv) The semigroup (e −tA ) t≥0 on E is of contraction type. 1(v) A has the bounded imaginary power (briefly BIP) property, i.e., there exist constants (2.1) 2(i) There exists a separable Banach space X such that the embedding E ⊂ X is dense and continuous. 2(ii) The linear map A has a unique extension to X. This extension map is still denoted by A and satisfies Assumptions 1(ii), 1(iii) and 1(iv).

Notation 2
For any γ > 0, the completion of E with respect to the norm |A −γ ·| will be denoted by D(A −γ ). For any γ > 0, the domain of the fractional power operator A γ (in E) will be denoted by D(A γ ). With few exceptions we will only speak about fractional powers of the operator A with respect to the space E and not X nor B and hence the notation A γ and D(A γ ) should be unambiguous. Those few exceptions are when we use notation D(A θ Y ), for instance in Theorem 3.2. Finally, let us note that since by assumption A −1 exists and is bounded (on E), the fractional powers A −γ , γ ≥ 0, are bounded (on E) too.
Before we proceed further we make the following useful remark.

Remark 2.1
Since E is a separable, UMD and martingale type p Banach space, we infer from [9, Remark 4.2, also Theorem A.4] that for every β ∈ R, the space B 0 = D(A β ) is also a UMD, martingale type p Banach space. The linear map A has an extension (or restriction depending on whether β is smaller or larger than 0) A 0 (usually denoted by A) to B 0 which satisfies Assumptions 1(ii), 1(iii) and 1(v). The operator −A 0 generates a contraction type semigroup which will still be denoted by e −tA t≥0 on B 0 . Moreover, if A has the BIP property then so has A 0 .
One of the consequences of the BIP property in Assumption 1(v) is that the fractional domains of the operator A are equal to the complex interpolation space of an appropriate order between D(A) and E, see e.g. [72].
Finally, if a linear operator A on a Banach space E is positive, then −A is the infinitesimal generator of a C 0 -semigroup in E, see for instance [10,Remark 2.1].
If E is a separable Banach space and q ∈ [1, ∞), we denote by L q (0, T ; E) (see, for instance, [29]) the Lebesgue space consisting of (equivalence classes of) Lebesgue measurable functions u : [0, T ] → E such that is finite. The space L q (0, T ; E) and W α,q (0, T ; E) equipped with the norms (2.2) and, respectively, are separable Banach spaces. We will denote by C([0, T ]; E) the Banach space of all E-valued continuous functions defined on the interval [0, T ] equipped with the supremum norm. Similarly, if k ∈ N * , then by C k ([0, T ]; E) we will denote the Banach space of all E-valued functions of C k -class. We will denote by C β ([0, T ]; E), for β ∈ (0, 1), a (non-separable) Banach space of all functions u ∈ C([0, T ]; E) such that We will also denote by H 1,q (0, T ; E) the space of functions u ∈ L q (0, T ; E) with weak derivatives Bu := u ∈ L q (0, T ; E). Endowed with the graph norm of B, the space H 1,q (0, T ; E) is a Banach space. By H 1,q 0 (0, T ; E) we denote the subspace of H 1,q (0, T ; E) consisting of all u ∈ H 1,q (0, T ; E) such that u(0) = 0. Now we introduce an important operator = T which will play a crucial role in our analysis. We start by setting Define also a linear operator A by the formula (2.7) The domain D(A) of A is a Banach space with norm Let us note that if A + κI , κ ≥ 0, satisfies parts 1(ii), 1(iii) and 1(v) of Assumption 1, then A + κI satisfies them as well, see Dore and Venni [33]. Finally, we define the operator by The domain D( ) is endowed with the graph norm., i.e, We recall that although is the sum of the closed operators A and B, it is not necessarily a closed operator. However, if E is an UMD Banach space and A + κI , for some κ ≥ 0, satisfies parts 1(ii), 1(iii) and 1(v) of Assumption 1 then, since = B − κI + A + κI , by Dore and Venni [33], see also Giga and Sohr [39], is a positive operator. In particular, has a bounded inverse. Consequently, one can define the fractional powers −α , α ≥ 0. In particular, for α ∈ [0, 1], −α is a bounded linear map in L q (0, T ; E), and for α ∈ (0, 1), for any t ∈ (0, T ), f ∈ L q (0, T ; E). Under the parts 1(i)-1(v) of Assumption 1, the parabolic operator and its fractional powers −α , α ∈ [0, 1], enjoy several nice properties for which we refer the reader to [10]. The properties of −α which are the most relevant to our study are summarized in the following lemma whose proof can be found in [10, Theorem 2.6 and Corollary 2.8].

Lemma 2.2
Assume that E is an UMD Banach space and an operator A satisfying Assumptions 1(ii)-1(iii) is such that A + κI , for some κ ≥ 0, satisfies Assumption 1(v). We also suppose that (A + κI ) −1 is a compact operator in E. Let α, β, δ be nonnegative numbers satisfying

Stochastic Preliminaries
The aim of this subsection is to introduce some additional probabilistic notation. We also present some basic results about stochastic integration with respect to compensated Poisson random measure.
We assume that P = , F, F, P , where F = F t ) t≥0 , is a filtered probability space satisfying the so called usual conditions, i.e., (i) P is complete on ( , F), (ii) for each t ∈ R + , F t contains all (F , P)-null sets, (iii) the filtration F is right-continuous.
Let us start by recalling the following definition which is taken from [45, Definition I.8.1].

Definition 2.3
In the framework of Assumption 2, a time-homogeneous Poisson random measure on (Z, Z) over P with the intensity measure ν ⊗ Leb, is a random variable satisfying the following conditions are pairwise disjoint, then the random variables η(U j ), j = 1, · · · , n are pairwise independent; (c) for all U ∈ Z and I ∈ B(R + ), is F-adapted and its increments are independent of the past, i.e., the increment between times t and s, t > s > 0, are independent form the σ -field F s .
If η is a time-homogenous Poisson random measure as above, then byη we will denote the corresponding compensated Poisson random measure defined bỹ with the convention that ∞ − ∞ = 0.
We proceed to the definition of functional spaces that we need throughout the paper. Suppose that Y is a separable Banach space. We denote by L q (Z, ν; Y ), q ∈ [1, ∞), the space of all (equivalence classes of) measurable functions ξ : Similarly, we define the space L p ( ; Y ) and L q ( T ; Y ), where T = [0, T ]× , see [29]. In the latter case, we consider the product σ -field B([0, T ]) × F. By L 0 ( ; Y ) we denote the set of measurable functions from ( , F) to Y .
For T ∈ (0, ∞] let N (0, T ; Y ) be the space of (equivalence classes of) progressively measurable processes ξ : Now for ξ ∈ M p step (0, T ; L p (Z, ν; E)) we set It is shown in [13] that if E is a Banach space of martingale type p ∈ (1, 2], thenĨ is a bounded linear map from M p step (0, T ; L p (Z, ν; E)) (with respect to the norm inherited from M p (0, T ; L p (Z, ν; E))) to L p ( , E). In particular, there exists a positive constant C > 0 which depends only on p and E such that (2.13) and EĨ (ξ) = 0 for any ξ ∈ M p step (0, T ; L p (Z, ν; E)). From these facts, we can define by Eq. 2.12 the stochastic integral of a process ξ ∈ M p step (0, T ; L p (Z, ν; E)) with respect to the compound random Poisson measureη. The extension of this integral to M p (0, T , L p (Z, ν, H )) is possible, thanks to the density of M p step (0, T , L p (Z, ν; E)) in the space M p (0, T , L p (Z, ν; E)). More precisely, we recall the following result whose proof can be found in [13, In particular, there exists a positive constant C which depends only on E and p such that for every ξ ∈ M p (0, T , L p (Z, ν; E)). Moreover, if ξ ∈ M p (0, T , L p (Z, ν; E)), then the process is an E-valued p-integrable martingale.
The above construction is done in the spirit of Métivier, see [56,Exercise 9,p. 195], see also [13] for further details. As pointed out to the authours by the referee, the construction of the Itô stochastic integral can also be done by defining the stochastic integral first for predictable integrands, and then extending the stochastic integral to progressively measurable integrands using the fact that the dual predictable projection of a Poisson random measure, see Theorem II.1.8 in [46], is absolutely continuous with respect to the Lebesgue measure (with respect to the "time" variable). See also [3] for a recent use of this classical notion.
As usual, we put Let us state the following useful result which can be proved using the argument of [ We close this section with the following maximal inequality whose proof can be found in [

Lévy Processes and Poisson Random Measures
The subject of this section is to give a short account on the correspondence between Lévy processes and Poisson random measures. Namely, given a Lévy process on a Banach space, one can construct a corresponding Poisson random measure. Conversely, given a Poisson random measure on a Banach space, one may find the corresponding Lévy process. To illustrate this fact, let us first recall the definition of a Lévy process.

Definition 2.7
Let Z be a Banach space. An Z-valued stochastic process L = {L(t) : t ≥ 0} over a probability space ( , F, P) is called an Z-valued Lévy process iff the following conditions are satisfied.
If F = {F t } t≥0 is a filtration on F, we say that an Z-valued stochastic process L = {L(t) : t ≥ 0} is a Lévy process over a filtered probability space ( , F, F, P) iff it is F-adapted, satisfies conditions (i), (iii-v), and (ii)' for all 0 ≤ s < t, the increment L(t) − L(s) is independent of F s .
In order to discuss Lévy processes in more detail we need to recall a definition of a Lévy measure.
is a characteristic function of a Borel 3 probability measure on Z.
The class of all Lévy measures on Z is denoted by L (Z).
For the readers convenience let us now list a few basic properties of Lévy measures, see [51], Theorem 5.4.8 (i,ii) and Proposition 5.4.5 (i,ii,iv). First let us recall a useful notation [51, p. 68]: x, a is a shortcut notation for the duality pairing x, a Z Z and, for r > 0, U r denotes the closed ball with radius r and centered at 0 in Z.

Proposition 2.9
Suppose that λ is a σ -finite Borel measure on a separable Banach space Z. Consider the following conditions.
Then we have the following implication: Suppose now that L = {L(t) : t ≥ 0} is an Z-valued Lévy process over a probability space ( , F, P), where Z is a separable Banach space. Then for each t ≥ 0 the measure μ t being the law of the Z-valued random variable L(t) is infinitely divisible and hence, see [51,Theorem 5.7.3], there exist a Lévy measure ν t on Z, a Gaussian measure ρ t on Z and a vector x t ∈ Z such that μ t = e s (ν t ) * ρ t * δ x t . (2.17) A proof of a finite dimensional version of this result can be found in [70,Theorem 8.1]. From now on we will assume that the process L is purely non-Gaussian, i.e., that ρ t = δ 0 for all t ≥ 0, see [70,Definition 8.2]. We also assume that x t = 0 for all t ≥ 0. Thus, see [51,Remark p Because L is a Lévy process, and not simply an additive process, the measures (μ t ) t≥0 form a convolution semigroup and therefore, the measure ν t is equal to tν, where ν = ν 1 . A purely non-Gaussian Z-valued Lévy process with x t = 0 for all t ≥ 0 satisfying (2.18) will be called a Lévy process with generating triplets 0, tν, 0 .
The following theorem is a generalisation of (a version of) Theorem 19.2 from [70]. The result below is an infinite dimensional generalisation of a summary of the first two steps of the proof of [ [50]. Finally, Lemma 20.5 in [70] is Theorem 1 on p. 29 in the Kahane's book [49]. Let 0 ∈ F be such that for every ω ∈ 0 , the function
Remark 2. 11 We have seen that since L is a Lévy process, ν t = tν. This implies as above that N is a time-homogenous Poisson random measure. Moreover, the equality (2.20) can be written as, The above theorem enables us to define an Itô integral with respect to the Lévy process L in terms of the corresponding compensated Poisson random measureÑ , where G is an appropriate process taking values in the space L(Z, V ), where V is an appropriate Banach space. From now we will consider only such processes that N restricted to (0, ∞) × Z \ U 1 is equal to 0.
Given a process G as above, we can define a process ξ In view of Section 2.2, the integral t 0 G(s)dL(s) is well defined for all t ∈ [0, T ], provided V is a martingale type p Banach space for some p ∈ (1, 2] and the process ξ defined above takes values in L 0 (Z, ν, V ) and belongs to the class M p (0, T , L p (Z, ν; V )). Moreover, for every q ∈ (1, p], there exists a constant C > 0 independent of G such that E sup In fact, using the above approach, we can extend the definition of the Itô integral with respect to the Lévy process to the integrands belonging to the whole class M p (0, T , L p (Z, ν; V )) by (2.26) We finish this section with a result somehow converse to Theorem 2.10 but whose proof, in the finite-dimensional case, can be traced to the Proof of Theorem 19.2 in [70,]. As we have observed earlier, this proof generalises to an infinite dimensional setting.
The details of the results presented in this subsection will be dealt in a separate publication [16].

Statements of the Main Results
In this section we will state our main results. For this purpose, we will introduce the problem, the concept of a martingale solution, and the main assumptions.
Let the Banach spaces E, X and the linear operator A be as in Assumption 1. We also assume that we have a probability space ( , F, P) and Poisson random measure η as in Assumption 2 and Definition 2.3. Throughout we fix T > 0.
We consider the following stochastic evolution equation: whereη is the compensated Poisson random measure corresponding to η, see Definition 2.3. Using the notation introduced in Notation 2, the assumptions on the nonlinear map G read as follows. Let us observe that this implies that the map A ρ− 1 In what follows, we will use the latter instead of G.
Next we will present assumptions on the drift operator F . For this purpose we first recall the notion of the subdifferential of the norm ϕ, for more detail see [23].
Given x, y ∈ X the map ϕ : R s → |x + sy| ∈ R is convex and therefore is right and left differentiable. Let us denote by D ± |x|y the right/left derivative of ϕ at 0. Then the subdifferential ∂|x| of |x|, x ∈ X, is defined by where X * is the dual space to X. One can show that not only ∂|x| is a nonempty, closed and convex set, but also ∂|x| = {x * ∈ X * : x, x * = |x| and |x * | ≤ 1}.
In particular, ∂|0| is the unit ball in X * .
Assumption 4 (i) The map F : [0, T ] × X → X is separately continuous. (ii) There exist numbers k 0 > 0, q > 1 and k ≥ 0 such that with a(r) = k 0 (1 + r q ), r ≥ 0, the following condition holds for t ∈ [0, T ] , (iii) There exists a sequence (F n ) n∈N of bounded separately continuous maps from [0, T ] × X to X such that (a) F n satisfies condition (ii) above uniformly in n, With all the notations and concepts presented above we are ready to define a martingale solution to the problem (3.1). Let us add a remark that is surely obvious to many readers while leaving it out could lead to a confusion for some other readers. Although in order to present the problem we have used a probability space and a PRM, these two objects are part of the solution. The only given objects are the space (Z, Z) and measure ν ∈ M + (Z). Definition 3.1 Let us assume that E and X are Banach spaces satisfying parts 1(i), 2(i) and 2(ii) of Assumption 1. Let us also assume that ν is a σ -finite nonnegative measure on a measurable space (Z, Z), i.e., ν ∈ M + (Z). Let p ∈ (1, 2] a real number as in part 1(i) of Assumption 1.
An X-valued martingale solution to the problem (3.1) is a system such that (i) ( , F, F, P) is a complete filtered probability space with a filtration F = {F t : t ∈ [0, T ]} satisfying the usual conditions, (ii) η is a time-homogeneous Poisson random measure on (Z, B(Z)) with intensity measure ν ⊗ Leb over ( , F, F, P), and for any t ∈ [0, T ], P a.s., If in addition there exists a separable Banach space B such that then the system (3.3) will be called an X-valued martingale solution to problem (3.1) with càdlàg paths in B.
We will say that the X-valued martingale solution to problem (3.1) with càdlàg paths in B is unique iff for any other martingale solution to Eq We refer to a recent paper [15] where the uniqueness in law of processes defined by stochastic convolutions with respect to PRM's are discussed.
We we formulate our main theorem. Let q be the number from Assumption 4(ii). Assume that q < q max , where Assume also that there exists a separable UMD Banach space Y such that A has an extension A Y which satisfies the parts and D(A θ Y ) ⊂ X for some θ ≤ 1 − q q max . Then, for any u 0 ∈ X problem (3.1) has an X-valued martingale solution with càdlàg paths Moreover, for anyq ∈ (q, q max ) and r ∈ (1, p) the stochastic process u satisfies are Lipschitz continuous uniformly with respect to t ∈ [0, T ], i.e., there exists K > 0 such that for all t ∈ [0, T ] and all u 1 , u 2 ∈ X, then the SPDEs (3.1) has a unique strong solution. In our work we are interested in the case when both these conditions are relaxed.
In order to prove Theorem 3.2 we will consider an auxiliary problem for which we will prove an auxiliary existence result (see Theorem 3.4) which holds under more restrictive conditions than the ones stated above. More precisely, we require that the nonlinear maps F and G satisfy the following set of conditions.
are bounded and separately continuous.

Remark
The above assumption can also be stated in a more precise way, see Assumption 3.
To be precise, we could request that there exists ρ ∈ [0, 1 p ) such that the maps are bounded and separately continuous. We state the following theorem whose proof will be given in Section 8. Although it is only an auxiliary result, it is still important as it is the main tool for the proof of Theorem 3.2 and we are not aware of a related result in the existing literature.

Theorem 3.4 Let E and A be as in Assumption 1 and (Z, Z, ν) be a measure space with ν ∈ M + (Z). Let the hypothesis in Assumption 5 be satisfied. Then, for every
Moreover, the stochastic process u satisfies (3.12) In view of Section 2.3, Theorem 3.4 can be written in terms of a Lévy process as follows.

Formulation of our Results in Terms of Lévy Processes
Let Z be a separable Banach space with the Borel σ -field Z = B(Z). Assume that L = {L(t) : t ≥ 0} is a Z-valued Lévy process with generating triple (δ 0 , tν, 0), see Theorem 2.10, such that for some fixed p ∈ (1, 2], where as in Section 2.3, U 1 is the closed unit ball in Z. Note that if supp ν ⊂ U 1 , then the Poisson random measure N corresponding to L restricted to (0, ∞) × Z \ U 1 is equal to 0.
Here, instead of Assumption 3 we assume the following set of hypotheses.

Assumption 6
There exists ρ ∈ (0, 1 p ) such that the diffusion coefficient G is bounded and separately continuous map (3.14) Remark Let us notice that this framework is less general than the one of Poisson random measures. In particular, if the map G satisfies Assumption 6 then the map G defined by satisfies, in view of Eq. 3.13 and the continuous embedding E ⊂ X, Assumption 3. We consider the following stochastic evolution equation In view of above remark and Section 2.3 we get the following result which is crucial in our reformulation of Theorem 3.4 in terms of Lévy processes. Theorem 3.5 Assume that the Banach space E, the linear map A and the map F satisfy the assumptions of Theorem 3.4. Let us assume that Z is a separable Banach space and Y = Y (t) : t ≥ 0 is an Z-valued Lévy process defined on a probability space ( 0 , F 0 , P 0 ) with the system of generating triplets δ 0 , tν, 0 such that supp ν ⊂ U 1 and, for some p ∈ (1, 2], the condition (3.13) is satisfied. Assume that the map G satisfies Assumption 6.
Proof of Theorem 3.5 This result readily follows from Theorem 3.4 because of the following argument. Let us consider a separable Banach space Z and an Z-valued Lévy process Y = Y (t) : t ≥ 0 , defined on a probability space ( 0 , F 0 , P 0 ) with the system of generating triplets δ 0 , tν, 0 such that supp ν ⊂ U 1 , and condition (3.13) is satisfied for some p ∈ (1,2]. Let N be the corrsponding Poisson random measure given by Theorem 2.10.
Since the map G satisfies Assumption 6, by Theorem 3.4, there exists a system In particular, η is a time-homogeneous Poisson random measure on a Banach space Z with the intensity measure ν ⊗ Leb such that condition (3.13) is satisfied. Applying Theorem 2.12 we can find a Z-valued Lévy process L = {L(t) : t ≥ 0} with the generating triplets δ 0 , tν, 0 . By the results discussed in Section 2.3 we infer that the system is a martingale solution to problem (3.15).

Outline of the Proof of Theorems 3.2 and 3.4
The detail of the proofs of the theorems 3.2 and 3.4 are given in Sections 9 and 8, respectively. These proofs are very technical and to make the reading of the paper easy, we outline the proofs of Theorem 3.4 and Theorem 3.2 in this subsection.

Outline of the proof of Theorem 3.4
The proof relies on a combination of approximation and compactness methods. Namely, we approximate the initial condition u 0 by a sequence (x n ) n∈N ⊂ E satisfying as n → ∞. We also define a sequence (u n ) n∈N of adapted E-valued processes by whereû n is defined bŷ and φ n : is the integer part of t ∈ R. Here we have used the following shortcut notation where Leb denotes the Lebesgue measure. Let us point out that between the grid points, Eq. 3.18 is linear, therefore, u n is well defined for all n ∈ N. Secondly, we proved that for any α ∈ (0, ρ) and ρ ∈ (0, ρ) there exists a constant C such that the following inequalities hold The proofs of these uniform estimates are non-trivial and rely on Lemmata 7.2 and 7.4, Proposition E.1 and the maximal regularity for deterministic parabolic equations.
Thirdly, by defining a sequence of Poisson random measures (η n ) n∈N by putting η n = η for all n ∈ N, we will prove that for any ρ ∈ (0, ρ) the family of the laws of random variables ((u n , η n )) n∈N is tight on the cartesian product space . Because a cartesian product of two compact sets is compact, it is sufficient to consider the tightness of the components of the sequence ((u n , η n )) n∈N .
For this aim let us define two auxiliary sequences of stochastic processes by The tightness of laws of the processes (u n ) n∈N on L p (0, T ; E) ∩ D([0, T ]; D(A ρ −1 )) follows by observing that and using Lemma 2.2 (for −1 f n ) and the Lemmata 7.6 and 7.7 (for v n ). The tightness of the laws family of (η n ) n∈N follows from [62,Theorem 3.2]. This tightness result along with the Prokhorov theorem and the modified Skorohod Representation Theorem, see Theorem C.1, implies that there exist a probability space ,η n = η * , and L((û n ,η n )) = L((u n , η n )), for all n ∈ N. Taking the new filtrationF as the natural filtration of (û n ,η n , u * , η * ) we prove that over the filtered probability space (ˆ ,F ,F,P) the objectsη n and η * are time-homogeneous Poisson random measure with intensity measure ν ⊗ Leb. We also prove that in appropriate topologŷ where a processv n is defined analogously to the way we have defined the process v n by replacing u n withû n . Using (3.20) and the uniform a priori estimates we obtain earlier we can pass to the limit and derive thatP-a.s.
This ends the proof of Theorem 3.4.
The scheme of the proof of Theorem 3.2 is very similar to the above idea, but it is longer and more complicated. In the next paragraph we will simply outline the main ideas of the proof and refer the reader to Section 9 for more details.
Outline of the proof of Theorem 3.2 The proof of Theorem 3.2 also relies on approximation and compactness methods. We mainly exploit the Assumption 4(iii) to set in the Banach space E an approximating problem with bounded coefficients. This approximating (auxiliary) problem takes the form which, thanks to Theorem 3.4, has an E-valued martingale solution with càdlàg paths in We denote this martingale solution by ( n , F n , F n , P n , η n , u n ).
The stochastic process u n can be written in the form The first step of the proof is to derive uniform a priori estimates concerning the convolution processes z n and v n . In this step the results obtained in Section 7, especially Lemmata 7.9 and 10, will play an important role. In fact, thanks to parts 1(i)-1(v) of Assumption 1, we can apply these results and deduce that for anyq ∈ (q, q max ) and r ∈ (1, p), Let us fixq ∈ (q, q max ) and ρ ∈ (0, ρ). We set B 0 = D(A ρ −1 ) and put In the following step we will prove that the family of laws of (z n , v n , η n ) n∈N is tight on X T . For this aim we will prove that the laws of the sequence (v n ) n∈N , respectively (η n ) n∈N  Firstly we will need to use Lemma 9.1, see [23] for a proof, as well as the previous uniform estimates, to prove that for somep ∈ (1, q max q ) sup n∈N |F n (·, u n (·))| Lp(0,T ;Y ) is bounded in probability. Observing that z n = −1 F n (·, u n (·)) and using Lemma 2.2, which is applicable thanks to parts 1(i)-1(v) of Assumption 1, we will deduce that for some θ ≤ 1 − 1 p the family of laws of (z n ) n∈N is tight on C([0, T ]; D(A θ Y )) and hence on C([0, T ]; X). This along with the Prokhorov Theorem and the modified Skorohod Representation Theorem, see Theorem C.1, imply that there exist a probability space (ˆ ,F ,P) and X T -valued random variables (z * , v * , η * ), (ẑ n ,v n ,η n ), n ∈ N, such thatP-a.s. (3.23) and, for all n ∈ N,η n = η * and L((ẑ n ,v n ,η n )) = L((z n , v n , η n )).
Next we will carefully construct a new filtrationF and prove that over the new filtered probability space (ˆ ,F ,F,P) the objectsη n and η * are time-homogeneous Poisson random measure with intensity measure ν ⊗ Leb. We also prove that in appropriate topologŷ whereû n = e −·A u 0 +ẑ n +v n . Putting u * := e −·A u 0 + z * + v * and using (3.23) and the uniform a priori estimates we obtained earlier we can take the limit and deduce thatP-a.s.
This ends the hardest part of the proof of Theorem 3.2. The scheme of the proof of Theorem 3.4 is very similar to the above idea and simpler. We refer the reader to Sections 9 and 8 for the omitted details.

Application I: Reaction-diffusion Equations with Lévy Noise of the Spectral Type
Lévy process, i.e., a Lévy process with the Lévy measure ν α given by Our aim in this section is to study an equation of the following type, We will achieve our aim by finding the conditions on the coefficients so that Theorem 3.2 is applicable. For this purpose we will reformulate problem (4.2) using a more general setting and the language of the Poisson random measures. Firstly, we denote by X = C 0 (O) the space of real continuous functions on O which vanish on the boundary ∂O. For γ ∈ R and r ∈ (1, ∞) by the symbol H γ,r (O) we will denote the fractional order Sobolev space defined by mean of the complex interpolation method, see [ In the case β ∈ R + \ N, the fractional order Sobolev spaces H β,p (O) can be defined by the complex interpolation method, i.e., with the Dirichlet boundary conditions, i.e., Next, let us consider a separately continuous real valued functions f defined on [0, T ] × O × R satisfying the following condition. There exists a number K > 0 such that for It is not difficult to prove that if f satisfies (4.5), then Therefore, by [10, Proposition 6.2] the Nemytskii map F defined by satisfies items (i) and (ii) of Assumption 4 on X.
we obtain a sequence (F n ) n∈N defined by which satisfies Assumption 4(iii).
Next we reformulate the noise appearing in the problem that we want to study. In view of the results of Section 2.3, see [1], to the real-valued Lévy process L there corresponds a time-homogeneous Poisson random measure η with Lévy measure ν α on Z = R defined by Eq. 4.1. Moreover, the Lévy processes Now, let g be a bounded and separately continuous function defined on [0, T ] × O × R and taking values in R. Furthermore, we assume that g(t, x, ·) is continuous uniformly w.r.t. (t, x). Let G be defined by (4.7) With this notation problem (4.2) can be rewritten in the following form The following theorem is a corollary of Theorem 3.2.  Proof Let us fix parameters α, p, d, q as in the assumptions. Next let us choose real numbers r > max{qd, 2d} and κ ∈ ( d r , 1 q ). Let us also choose X = C 0 (O), E = H κ,r 0 and B = L r . Let us put δ = κ 2 . Then, since 1 2 ≤ 1 p we infer that δ < 1 2q < 1 2 and δ < 1 p . We also deduce that Next, we denote by A = A r the minus Laplace operator − with the Dirichlet boundary conditions in the space B. Since r ≥ p, r ≥ 2d and p ∈ (1, 2] we infer that E and B are separable, UMD and type p Banach spaces. Now it is well known that the assumptions of the first and second part of Theorem 3.2 are satisfied by A r . We put Z = R and ν = ν α . Then, we immediately see that We define a mapG bỹ The mapG may not be defined on the whole space X, but the map A −δ rG is because δ = κ 2 . Indeed, we have the following chain of inequalities.
Since the function g is continuous one can easily check that the continuity condition in Assumption 3 is satisfied. Observe that where G is defined in Eq. 4.7.
Since κ > d r we have E ⊂ X. Moreover, it is straightforward to check that the nonlinear map F defined by Eq. 4.6 satisfies Assumption 4 on X. Finally Thus all the assumptions (with our choice of spaces and maps) of Theorem 3.2 are satisfied and therefore the proof of the existence of a solution with the requested properties follows.

Application II: Reaction-diffusion Equations of an Arbitrary Order with Space-Time Lévy Noise
The aim of this section is to show how Theorem 3.4 can be applied to Stochastic reactiondiffusion equations driven by the space-time Poissonian white noise. This type of noise, which is a generalization of the space-time white noise, has been treated quite often in the literature, see for instance [65,Definition 7.24] (see also [7]).
We begin by introducing the assumptions on the drift of our problem and the driving noise. In the second subsection we will present the detail about the coefficient of the noise. Finally, after a careful statement of the assumptions on our problem we will formulate and prove the existence of a martingale solution to stochastic reaction-diffusion equations driven by a space-time Poissonian noise.

The Noise and the Deterministic Nonlinear Part of the Problem
Let d ≥ 1, p ∈ (1, 2], and O be a bounded open domain in R d with boundary ∂O of C ∞ class. We consider a complete filtered probability space (¯ ,F ,F,P) where the filtration F = (F t ) t≥0 satisfies the usual conditions.
The assumptions on the noise in the equation that we are interested in are given below.
We fix p as above for the remainder of this section. We also putν = Leb ⊗ν, where Leb is the Lebesgue measure on O.
Let also k be a positive integer. Borrowing the presentation of [10, Section 6.3] we introduce a differential operator A of order 2k as follows.
(a) The differential operator A defined by is properly elliptic (see [73,Section 4.9.1]). The coefficients a α are C ∞ functions on the closureŌ of O.
with the coefficients b j,α being C ∞ functions on ∂O. The orders m j of the operators B j are ordered in the following way: We assume that m k < 2k and where n ξ is the unit outer normal vector to ∂O at ξ ∈ ∂O. (c) For any x ∈Ō and ξ ∈ R n \{0} let a(x, ξ ) = |α|=2k a α (x)ξ α . We assume that Here T x (O) is the set of all tangent vectors to ∂O at x ∈ ∂O and τ + j (t) are the roots with positive imaginary part of the polynomial defined by C τ → a(x, ξ + τ n x ) − t.
The differential operator A induces a linear unbounded map A r on the Banach space L r (O), r > 1, defined by We denote by F the Nemytskii map associated to f , i.e., defined by and assume that F : The restrictions of F to [0, T ] × H θ,r B will also be denoted by F .

Coefficient of the Noise
We begin this subsection with the precise statement of the assumptions on the coefficient of the noise.
Indeed, is linear and by Corollary B.4 and Eq. 5.1 we have the following chain of equalities/inequalities Finally, by the choice of θ , r, and p above, the embeddings are continuous, so we can define a nonlinear map G by In what follows we will also denote by G the restriction of the previously defined map G to the sets [0, T ] × H θ,r B with r ∈ (p, ∞) and θ ≥ 0. It follows from the corresponding properties of the map G 0 that for every t ≥ 0, G(t, ·) is continuous and that for each u ∈ H θ,r B the function G(·, u) is strongly measurable.

The Formulation of the Result
Letη be the compensated Poisson random measure associated to the time-homogeneous Poisson random measure given by Remark 5.1. With the functional setting we described above, the problem that we are interested in is (5.14) Remark 5.3 A very important example of problem (5.14) is the following SPDE where A is a second order differential operator, both f and g are continuous and bounded real functions defined on R and, roughly speaking,L denotes the Radon-Nikodym derivative of the space-time Lévy white noise 5 L, i.e., We are finally ready to define the concept of solution to problem (5.14).  (iv) u is a L r (O)-valued càdlàg process. 5 We refer again to Appendix A for the definition of space-time Lévy noise. 6 We refer to Appendix A for the definitions and facts about space-time Lévy and Poissonian noise (v) for every t ∈ [0, T ], u satisfies the following equation P-a.s. The above theorem can be reformulated in terms of space-time Lévy noise, but since such a result would not be significantly different from the last one, we omit it and leave as an exercise to an interested reader.
Proof of Theorem 5.6 Let us fix the numbers d, k, p, and r, the space E and the operator A r as in the statement of the theorem. Also, let F (resp. G) be defined by equality (5.9) (resp. (5.12)).
Since r ≥ p, the separable Banach spaces E and B are UMD and martingale type p. As we mentioned above A r has the BIP property on E, is a positive operator with compact resolvent and −A r generates a contraction type C 0 -semigroup on E. Owing to Claim 5.
satisfies Assumption 5. Hence, since Theorem 3.4 is applicable, we infer that problem (5.14) has a E-valued martingale solution u. Since A r is a infinitesimal generator of a contraction C 0 -semigroup on L r (O), by [77], the paths of the process u are càdlàg in L r (O).

Remark 5.7
Let O be a bounded open domain in R d , with d ≥ 1. Let n be a fixed natural number. For each i = 1, · · · , n let ν i be a Lévy measure on R satisfying (5.1). For each i = 1, · · · , n, let {L i (t); t ≥ 0} be a Lévy noise with Lévy measure ν i . For a fixed T ∈ (0, ∞) we consider the following system of SPDEs , u 1 (t, x), . . . , u n (t, x))dL (t) +f i (t, x, u 1 (t, x), . . . , u n (t, x) n, are differential operators of order 2k satisfying conditions (a)-(d), see pages 27 and 28. Furthermore, we assume that are separately continuous and bounded. In addition, we assume that g(t, x, ·) is continuous uniformly w.r. t. (t, x). Problem (5.19) was studied by Cerrai in [20] when each L is a Wiener process.
We will apply the previous theorem on the Banach space The existence result we claimed earlier is now a straightforward consequence of Theorem 5.6.

Application III: Stochastic Evolution Equations with Fractional Generator and Polynomial Nonlinearities
In this section we will deal with a problem that is similar to the problem from the previous section, but with one important modification. From now on we will assume that the nonlinear term F is of polynomial growth. We put k = 1 and assume that A and B are differential operators satisfying conditions (a)-(d), see page 27. As in the previous section we fix r > 1 and denote by A r be the linear operator induced by A in the Banach space L r (O).
We also consider a separately continuous function f : [0, T ] × O × R → R satisfying condition (4.5), for some q ≥ 1. We denote by F the Nemytskii operator defined by  With the various mappings we have introduced above we consider the following SPDEs In addition to the assumptions on G above we also assume that pd < 2kγ and that F satisfies Assumption 4.5 with some q ∈ (1, p). If r > max{p, pd p−q }, then for any u 0 ∈ C 0 (O) there exists a C 0 (O)-valued martingale solution to Eq. 6.5 with càdlàg trajectories in L r (O).
Before we embark on the proof of this result let us make the following remark.

Remark 6.3
If the intensity measure ν of the space-time white noise is finite, then as in the proof of Theorem IV.9.1 in [45] the solution can be written as a concatenation of solutions to the deterministic reaction-diffusion equations on random intervals with the initial data being a measure-valued random variable.
To be more precise, let λ := Leb(O) × ν(R), (τ i ) i∈N be a family of independent, exponentially distributed random variables with parameter λ and Let also (Y i ) i∈N be a family of independent ν/ν(R) distributed random variables and {x i : i ∈ N} be a sequence of independent and uniformly distributed random variables in O. Then, the space-time white noise η can be written as follows: for Using this representation the above SPDEs can be described by a deterministic PDE with initial condition being a measure in the time intervals [T n , T n+1 ), i.e., u solves the deterministic PDE ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ∂ ∂t u(t, ξ ) + Au(t) dt = f (u(t, ξ )) ξ ∈ O, t ∈ (T n , T n+1 ), u(T + n , ξ) = u(T − n ) + Y n δ x n , ξ ∈ O, u(t, ξ ) = 0, for ξ ∈ ∂O, t ∈ (T n , T n+1 ).

(6.6)
It follows that our conditions have to be stronger than the conditions in [8], which is indeed the case. In fact, for γ = 1 we assume that d < 2 q which is stronger than d ≤ 2 q−1 imposed by Brezis and Friedman in [8].
Proof of Theorem 6.2 We just give a sketch of the proof because it is very similar to the proofs of Theorem 4.1 and Theorem 5.6. Let us fix the numbers d, γ , p, q, and r as in the statement of the theorem. We denote by A γ r the fractional power of the linear operator A r induced by −A on the Banach space B = L r (O). Also, let F be defined by equality (6.3).
By Remark 5.2 we can find θ > d r such that θ + d − d r < 2kγ p and the Banach space E = H θ,r B is continuously embedded in X := C 0 (O). Thus, owing to the assumption on g (resp. f ) we can argue as in the proof of Claim 5.1 (resp. Claim 6.3) to prove that the map G (resp. F ) satisfies Assumption 3 (resp. Assumption 4) with ρ ∈ (0, 1 p − d 2kγ ). Since r ≥ p, E and B are separable, UMD and type p Banach spaces. Finally, let Y = L r (O) and A Y = A γ r . Since 1 − q q max > 1 − q p and r > pd p−q , we can find κ 1 ∈ ( d r , 1 − q p ) such that Since −A γ r is an infinitesimal generator of a contraction type C 0 -semigroup on Y = B = L r (O), all the assumptions of Theorem 3.2 are satisfied by problem (5.14). Hence, we easily conclude the proof of Theorem 6.2 from the applicability of Theorem 3.2.

Some Preliminary Results about Stochastic Convolution
In this section we will state several results concerning the stochastic convolution process.

The Stochastic Convolution
Let us begin with listing the assumptions we will be using throughout the whole section. We assume that E and A are respectively a Banach space and a linear operator satisfying parts 1(i)-1(v) of Assumption 1. A real number p ∈ (1, 2] satisfies part 1(i) of Assumption 1 and ρ ∈ (0, 1 p ) satisfies Assumption 3. We also assume that the following are given: a measurable space (Z, Z), a nonnegative measure ν ∈ M + (Z) on (Z, Z), a filtered probability space P = ( , F, F, P) such that the right-continuous filtration F = (F t ) t≥0 satisfies the usual conditions, and a timehomogeneous Poisson random measure η with Lévy measure ν. For any progressively measurable process ξ : [0, ∞) × → L p (Z, ν; E) such that one can define the so called stochastic convolution process by the following formula   ∈ (0, ρ), then there exists a constant C 1 > 0 such that Proof of Lemma 7.2 Let us fix ρ ∈ (0, ρ) and put δ = ρ + 1 p − ρ ∈ (0, 1 p ). Let us also choose ξ ∈ B M (E) and put u = S(ξ ). Since, by [9], a type p UMD Banach space is a martingale type p Banach space, by Eq. 7.2 and Theorem 2.4 we infer that the stochastic convolution process and δ = ρ + 1 p −ρ we infer that u belongs to L p (0, T ; D(A ρ )) almost surely. Furthermore, Since pδ ∈ (0, 1), it follows from Eq. 7.5 and the Fubini theorem that Thus the proof of Lemma 7.2 is complete.

Lemma 7.3 Let the assumptions of Lemma (7.2) hold. Let ρ ∈ (0, ρ) and put
(ii) and the process u = S(ξ ) admits a B-valued càdlàg modification (which will be still denoted by S(ξ )). Let δ := 1 p + ρ − ρ < 1 and β := 1 − 1 p + ρ − ρ > 0. By applying A δ to both sides of the identity (7.6) and by noticing that Using the inequality (2.14) in Theorem 2.4 we obtain Next applying Hölder's inequality twice and invoking inequality (7.5) we get This completes the proof of (i) with C 2 (T ) = CC 21  is continuous, in view of the identity (7.7) and Theorem 2.4 we easily deduce that the process u has a D(A ρ−1 )-valued càdlàg modification. This completes the proof of part (ii) of our lemma. Since, by [9], any UMD Banach space of type p is also an martingale type p Banach space, and ξ satisfies (7.2), part (iii) is easily deduced by applying [77, Corollary 5.1].
The next lemma is about estimates of S(ξ ), ξ ∈ B M (E) in the Besov-Slobodetskii spaces W α,p (0, T ; E), see its definition on page 5. α ∈ (0, ρ). Then there exists a number C 3 > 0 such that

Lemma 7.4 Let the assumptions of parts (i)-(ii) of Lemma 7.3 hold. Assume that
Proof Let us fix ξ ∈ B M (E) and put u = S(ξ ). Let us fix α ∈ (0, ρ) and let us choose an auxiliary ρ ∈ (α, ρ). In view of Lemma 7.2 and the definition of the (2.3) it is sufficient to estimate the mathematical expectation of the seminorm (2.3) of u. For this aim, without loss of generality, we can take s < t ∈ [0, T ]. As in the Gaussian case we have In view of the definition (2.3) it is sufficient to prove that there exist two positive numbers C 31 , C 32 such that Let us begin with estimating S 1 . By using the Fubini Theorem, [13, Corollary C.2], the and the definition (7.2) of the class B M (E), we infer that In order to study the term S 2 let us recall, see [63,Theorem II.6.13], that there exists a C > 0 such that ≤ C h γ , h > 0. (7.8) Therefore, by applying the Young inequality for convolutions we infer that . Invoking Lemma 7.2 and the estimate for S 1 concludes the proof of the lemma.

Remark 7.5
Since α ∈ (0, 1 p ) we cannot infer from the above Lemma that the process S(ξ ), ξ ∈ B M (E), has an E-valued càdlàg modification. It is known, see for instance [48], that if E is a Hilbert space, the driving Lévy process L lives in E and {e −tA : t ≥ 0} is contraction type C 0 -semigroup on E, then the stochastic convolution process has an E-valued càdlàg modification. If E is a Banach space, then it is sufficient to assume that either E is a p-smooth Banach space or the semigroup {e −tA ; t ≥ 0} on E is analytic, see [77]. However, we should note that in our framework we do not have such a nice situation. Indeed, roughly speaking, our semigroup is analytic and contractive on a martingale type p, p ∈ (1, 2], Banach space E and our noise lives in a larger space than E (say D(A −α ), α > 0), and in general even if −A is the infinitesimal generator of an analytic semigroup of contraction type it is not known whether the stochastic convolution S(ξ ), ξ ∈ M p (0, T , L p (Z, ν; E)), has a càdlàg modification in E or in a smaller space, say D(A γ ), γ > 0, than E. This is even an open question for the case when E is a Hilbert space, see for instance [66]. For the proof of this lemma we need the following general result. Let ε > 0 and K ε = {y ∈ X : |y| ≤ (Cε −1 ) 1 p M}. It follows easily from the Chebyshev inequality that Corollary D.2-(a) follows from this inequality and the compactness of the embedding E ⊂ Y .
Next, let us fix 0 ≤ σ ≤ τ ≤ T . Then by [13,Corollary C.2] and the Jensen inequality Thus we can apply Corollary D.2-(b) from which the sought result follows.
Let us fix ξ ∈ B M (E) and put u = S(ξ ). Then, by the Hölder and Jensen inequalities, where Lemma 7.2 and Lemma 7.3-(iii) were used to obtain the last inequality. The proof of the first part is complete.
To prove the second part we observe that by the same argument as above, given q and r, we can find ε > 0 and C > 0 such that Moreover, by Lemma 7.4, for any fixed α ∈ (0, ρ), we can find C 3 > 0 such that is compact. Hence the second part of the Lemma follows.  (0, ρ). Then, for every q ∈ (p, p 1−ρ ) and every r ∈ (1, p), there exists C > 0 such that E|S(ξ )| r L q (0,T ;E) ≤ CM r , ξ ∈ B M (E). (7.11) Proof of Lemma 10 Let q ∈ (p, p 1−ρ ), ρ ∈ (0, ρ) and define θ = p q . As in proof of Lemma 7.9 we let Y = D(A ρ ) and B = D(A ρ −1 ). Then by the reiteration property of the complex interpolation we have the continuous embedding,

Lemma 7.10 Let the assumptions of parts (i) and (ii) of Lemma 7.3 be satisfied with ρ ∈
Owing to Lemma 7.2 and parts (i) and (ii) of Lemma 7.3 we can argue exactly as in the proof of Lemma 7.9 and show that for any r ∈ (1, p) we have (7.12) which implies inequality (7.11). Thanks to Eq. 7.12 we can again use the same argument as in proof of Lemma 7.9 to deduce that the family of the laws of {S(ξ ) : ξ ∈ B M (E)} on L q (0, T ; E) is tight.

Proof of Theorem 3.4
We begin the proof of Theorem 3.4 by introducing a sequence of approximating processes. Let us fix for the whole section a number T > 0. Consider a sequence (x n ) n∈N is the integer part of t ∈ R. Let us define a sequence (u n ) n∈N of adapted E-valued processes by u n (t) = e −tA x n + whereû n is defined bŷ and where we have used the following shorthand notation (Here Leb denotes the Lebesgue measure.) The E-valued processû is piecewise constant adapted and hence progressively measurable. Between the grid points, Eq. 8.1 is linear, therefore, u n is well defined for all n ∈ N.
Next we will prove certain uniform estimates for the sequence (u n ) n∈N . Let us recall that F is a bounded nonlinear map defined on [0, T ] × E and taking values in D(A ρ−1 ). Furthermore, it is separately continuous.
Note that, by definition, u k n is equal to the restriction of u n to J k and u n = u 2 n −1 n . Hence to prove our proposition it is sufficient to check that the estimates (8.3)-(8.5) are true and uniform w.r.t k on J k for u k n andû k n with k = 0, ..., 2 n − 1. On the interval J 0 we have Hence combining these two remarks with Lemma 7.2, Lemma 7.4 and Proposition E.1 we infer that (8.3)-(8.4) are true on J 0 for u 0 n andû 0 n . Using the same approach we can prove by induction that for each α and ρ as above there exists a constant C > 0 such that for any n ∈ N and k ∈ {0, ..., 2 n − 1} we have With the same argument as above we check that (8.3)-(8.5) are correct and uniform w.r.t k on each J k with k = 0, ..., 2 n − 1. With this fact and the identity u n = u 2 n −1 n , we conclude the proof of our proposition.
We will also need the following result.
Then, the sequence g n n∈N defined by Eq. 8.6 is convergent (and hence the set {g n : n ∈ N} from which we derive the convergence in L p (0, T ; E).
After these preliminary claims we are now ready for the proof of Theorem 3.4 which will be divided into several steps. But before we go further let us define a sequence of Poisson random measures {η n } n∈N by putting η n = η for all n ∈ N.
Step (I) The family of the laws of ((u n , η n )) n∈N is tight on , for any ρ ∈ (0, ρ).
Proof To simplify notation we set B 0 = D(A ρ −1 ) for any ρ ∈ (0, ρ). Define three functions f n , g n and v n by We argue exactly as in [10]. We recall that the space M p (0, T ; E), the operators and A are defined on page 9 and 7, respectively. Since, by estimates ( Remark Let us observe that the space Z T defined above in Eq. 8.11 differs from the space X T defined earlier in Eq. 3.22. From Step (I) and Prokhorov Theorem (see, for instance, [24, Theorem 2.3]) we deduce that there exist a subsequence of ((u n , η n )) n∈N , still denoted by ((u n , η n )) n∈N , and a Borel probability measure μ * on Z T such that L(u n , η n ) → μ * weakly. By Theorem C.1 there exist a probability space (¯ ,F ,P) and a sequence ū n ,η n n∈N , of Z T -valued random variables such that the laws of ū n ,η n and u n , η n on Z T are equal, (8.12) and there exists a Z T -valued random variable (u * , η * ) on (¯ ,F ,P) with L((u * , η * )) = μ * , such thatP-a.s.
(ū n ,η n ) → (u * , η * ) in Z T , (8.13) andη n = η * for all n ∈ N. The sequence (ū n ) n∈N has similar properties as the original sequence (u n ) n∈N . Those we will use are stated in part (i) of the next step.
Step (II) The following holds Let us fix r ∈ (1, p). Since u * is Z T -valued, it follows from part (i) that the sequence ū n − u * r L p (0,T ;E) isP-uniformly integrable. Since by Eqs. 8.13 and 8.11, ū n − u * r L p (0,T ;E) → 0 on¯ , by applying the Vitali Convergence Theorem we deduce part (ii).
Before we continue we should note that the random variables u n , u * :¯ → L p (0, T ; E), induce two E-valued stochastic processes still denoted with the same symbols, see for example [18,Proposition B.4] for a proof for the space L ∞ loc (R + ; L 2 loc (R d )). Now letF = (F t ) t≥0 be the filtration defined bȳ 14) where N denotes the set of null sets ofF . Sinceη n = η * , it is easy to show that the filtration obtained by replacingη n with η * in Eq. 8.14 is equal toF. The next two steps imply that the following two E-valued integrals over the filtered probability space (¯ ,F ,F,P) t 0 Z e −(t−s)A G(s,ū n (s), z))η n (dz, ds), t ≥ 0, Step (III) The following holds Proof Before embarking on the proof, let us first recall that in view of Theorem C.1 we infer thatη n (ω) = η * (ω) for allω ∈¯ and n ∈ N.

Lemma 8.3
Assume that ( , F, P) is a probability space and Y is a Banach space and that (y n ) n∈N is a sequence of Y -valued random variables over ( , F, P) such that y n → y * weakly, i.e., for all φ ∈ Y * , Ee i φ,y n → Ee i φ,y * . If z is a another Y -valued random variable over ( , F, P) such that y n and z are independent for all n ≥ 1, then y * and z are also independent.
Proof of Lemma 8.3 The random variables y * and z are independent iff The weak convergence and the independence of z and y n for all n ∈ N justify the following chain of equalities.

Proof of Step (III)-(ii)
We have to show that η * is a time-homogeneous Poisson random measure with intensity ν ⊗ Leb. But this will follow from Step (III)-(i), since η * (ω) = η m (ω) for all ω ∈ and m ∈ N.
Step (IV) The following holds (i) for every n ∈ N, the processū n is aF-progressively measurable; (ii) the E-valued process u * isF -progressively measurable.
Proof One may suspect that there is a simpler proof by by adaptiveness and left-continuity. However, here the problem is that u n ∈ D([0, T ]; Y ) with E ⊂ Y densely and continuously. Because G is only defined on [0, T ] × E, we want u as an E-valued process to be progressively measurable.
As we noted earlier, one can argue as in [18,Proposition B.4] and prove that the random variablesū n , u * :¯ → L p (0, T ; E) induce two E-valued stochastic processes still denoted with the same symbols. Here, we have to show that for each n ∈ N,ū n and u * areFprogressively measurable. By definition ofF, for fixed n ∈ N the processū n is adapted tō F by the definition ofF. Let us fix r ∈ (1, p). By Step (II) the processū n is bounded in L r (¯ T ; E), hence, there exists a sequence of simple functions (ū m n ) m∈N such thatū m n →ū n as m → ∞ in L r (¯ T ; E). In particularly, by using the shifted Haar projections used in [15,Appendix B] we can choose (ū m n ) m∈N to be progressively measurable. It follows thatū n is progressively measurable as a L r (¯ T ; E)-limit of a sequence of progressively processes. Finally, sinceū n → u * as n → ∞ also in L r (¯ T ; E), it follows that u * is progressively measurable.
Let μ be a time-homogeneous Poisson random measure over (¯ ,F ,F,P) with intensity measure ν ⊗ Leb, v be an E-valued progressively measurable process, u 0 ∈ D(A ρ− 1 p ) and K be a nonlinear map defined by Here, as usual,μ denotes the compensated Poisson random measure of μ.
Step (V) For all t ∈ [0, T ] and n ∈ N we haveP-almost surelȳ u n (t) − K(x n ,û n ,η n )(t) = 0, whereû n is defined bŷ Proof First, let ρ ∈ (0, ρ) and ) and . Again for simplicity we set B 0 = D(A ρ −1 ). It is proved in [14] that the map G : is well defined, linear and bounded. Therefore, for any n ∈ N, the two triplets of random variables (u n , η n ,û n ) and (ū n ,η n ,û n ), whereû n = G(u n ) andû n = G(ū n ), have equal laws on X 1 T × X 2 T . Second, let us define processesz n andz n bỹ z n (t) := e −tA x n + Let us also define processesz n andz n by replacing (u n , η) and (û n , η) by (ū n ,η n ) and (û n ,η n ) in formula (8.19) and (8.20), respectively. Thanks to the continuity of the linear map G and Assumption 5, it follows from [15,Theorem 1] that the quintuples of random variables (u n , η n ,û n ,z n ,z n ) and (ū n ,η n ,û n ,z n ,z n ) have the same law on Z T ×X 1 T . Consequently (u n ,z n ) and (ū n ,z n ) have equal laws on R, where the continuous functional : Therefore, for any function Now let ε > 0 be arbitrary and let φ ε ∈ C b (R, R + ) be defined by It is easy to check that P (ū n ,z n ) ≥ ε ≤ ¯ 1 [ε,∞) ( (ū n ,z n ))dP + ¯ 1 [0,ε) ( (ū n ,z n )) (ū n ,z n ) ε dP =Ēφ ε ( (ū n ,z n )).
This implies that forP almost all t ∈ [0, T ] and almost surelyū n (t) =z n (t). Since two càdlàg functions which are equal almost all t ∈ [0, T ] must be equal for all t ∈ [0, T ], we derive that almost surelyū n = K(x n ,û n ,η n ), for all t ∈ [0, T ] .
Step (VI) We have Proof First, notice that sinceη n = η * for any n ∈ N, the convergence in Step (VI) is equivalent toĒ Observe that for any any n as n → ∞. Sinceū n = K(x n ,û n ,η * ), arguing as in the proof Proposition 8.1 we can show that u n ∈ W α,p (0, T , E)P-a.s., for any α ∈ (0, 1 p ). Hence by inequality (E.1) it follows that P-a.s. The continuity of F (see Assumption 5), the convergence (8.23) and the Lebesgue DCT imply S n 1 → 0 as n → ∞. In a similar way we will show that S n 2 → 0 as n → ∞. In particular, from Fubini Theorem and Theorem 7.1 as well as (8.13) we infer that (8.24) hence by the continuity of G, the convergence (8.23) and the Lebesgue DCT S n 2 → 0 as n → ∞.
To establish Theorem 3.4 we need to check the following claim.
Proof Let us fix r ∈ (1, p). From Steps (II) to (VI) we infer thatū n → u * in L r (¯ T ; E), u n = K(x n ,û n ,η n ) in L r (¯ T ; E), By the uniqueness of the limit, we infer that

Proof of Theorem 3.2
In this section we replace the boundedness assumption on F by the dissipativity of the drift −A + F . The spaces E, X are as in Assumption 1, and we recall that E ⊂ X ⊂ D(A ρ−1 ). Before we proceed let us state the following important consequence of Assumption 4. Then Before giving the proof of Theorem 3.2 let us notice that Assumption 4 implies that |F (t, y)| X ≤ a(|y| X ), t ≥ 0, y ∈ X. Let us denote this martingale solution by ( n , F n , P n , F n , η n , n ) .
We denote by E n the mathematical expectation on ( n , F n , P n ). In view of Theorem 3.4, for each n ∈ N, u n has càdlàg paths in D(A where v n (t) =  Step (I) Let q max be defined by Eq. 3.7. Then for anyq ∈ (q, q max ) and r ∈ (1, p), we have Step (I) follows from Lemma 7.9 and Lemma 10.
Step (II) For anyq ∈ (q, q max ) and r ∈ (1, p) defined in Step ( Hence, we proved the first part of Step (II). Note that the last inequality implies that sup n E n |z n | rq Lq (0,T ;X) < ∞. Before we proceed further, we recall that there exist θ < 1 − q q max and an UMD, type p and separable Banach space Y such that D(A θ Y ) ⊂ X ⊂ Y . To prove the second part we use the identity z n = −1 Y F n (s, u n (s)), where −1 Y = B + A Y with A Y being defined as in Eq. 2.6 by replacing A with A Y , and Remark 2.1 along with Lemma 2.2. But first we need to show that for somep ∈ (1, q max q ), the sequence |F n (·, u n (·))| Lp(0,T ;Y ) n∈N is bounded in probability. Let us fixp ∈ (1, q max q ). By Lemma 9.1 and the continuity of the embedding E ⊂ X we infer that |F n (s, u n (s))|p X ≤ C(1 + |e −sA u 0 |p X + |v n (s)|p q X + |z n (s)|p q X ). From this inequality we easily deduce that |F n (·, u n (·))|p Lp(0,T ;X) ≤ C(1 + |v n |p q Lq (0,T ;X) + |z n |p q L ∞ (0,T ;X) ). (9.10) Takingq =pq ∈ (q, q max ) and raising to the power r q 2 both sides of Eq. 9.10 implies that |F n (·, u n (·))| rp q 2 Lp(0,T ;X) ≤ C(1 + |v n | rq Lq (0,T ;X) + |z n | rq L ∞ (0,T ;X) ).

Remark 9.2 Let q be a number in the interval [p, ∞). It follows from
Step (I) and Step (II) that for anyq ∈ (q, q max ) and r ∈ (1, p) sup n≥1 E n u n rq Lq (0,T ;X) ≤ C.
By proving that sup t∈[0,T ] |u n (t)| X is uniformly bounded, which implies that τ n ↑ T almost surely as n → ∞, and then using (9.2) and Lemma 2.2 they could show that the laws of z n is tight on C([0, T ]; X). In our framework we know a priori that u n is only càdlàg in D(A ρ −1 ) for ρ ∈ (0, ρ), hence τ n will not be a well defined stopping time and we will not be able to show that sup t∈[0,T ] |u n (t)| X is uniformly bounded.
In order to use Theorem C.1 we also need the following.
Step Let B 0 as in Step (III). From Steps (I), (II), (III) and Prokhorov's theorem it follows that there exists a subsequence of ((z n , v n , η n )) n∈N , also denoted by ((z n , v n , η n )) n∈N and a Borel probability measure μ * on X T , where the space has been defined earlier in Eq. 3.22, such that the sequence of laws of (z n , v n , η n ) n∈N converges to μ * . Moreover, by Theorem C.1, there exists a probability space (ˆ ,F ,P) and X T -valued random variables (z * , v * , η * ), (ẑ n ,v n ,η n ), n ∈ N, such thatP-a.s., (ẑ n ,v n ,η n ) → (z * , v * , η * ) in X T (9.12) and, for all n ∈ N,η n = η * and L((ẑ n ,v n ,η n )) = L((z n , v n , η n )) on X T .
We define a filtrationF = (F t ) t∈[0,T ] on (ˆ ,F ) as the one generated by η * , z * , v * and the families {z n : n ∈ N} and {v n : n ∈ N}, that is, for t ∈ [0, T ], (9.13) where N denotes the set of null sets ofF . The next two steps imply that the following two Itô integrals over the filtered probability Step (IV) The following hold Step (V) The following holds (i) for all n ∈ N, the processesv n andẑ n areF-progressively measurable; (ii) the processes z * and v * areF-progressively measurable.
The proofs of Step (IV) and (V) are the same as the proofs of Step (IV) and (V) of Theorem 3.4. Also, as earlier in the proof of Theorem 3.4, in order to complete the proof of Theorem 3.2 we have to prove the following claim.
Step (VI) Let u * = e −·A u 0 + z * + v * and K be the mapping defined by K(u 0 , u, η)(t) = e −tA u 0 + (9.14) Proof Since the process u n = e −·A u 0 + z n + v n is a martingale solution of problem (9.3) and L((ẑ n ,v n ,η n )) = L((z n , v n , η n )), for all n ∈ N, we can argue as in Step (V) of the previous section and prove thatP-a.s. u n (t) = K n (u 0 ,û n ,η n )(t), Thanks to Eq. 9.16 and this last convergence we can prove, by a similar argument used as above, that With a similar argument we can also show that Lq (0,T ;B 0 ) = 0, (9.19) and Lq (0,T ;E) → 0. Lq (0,T ;X) = 0, which is correct because the embedding E ⊂ X is continuous. In the other hand, by Proposition 2.5 we have wherer = r 2qq and p * = min(q q , p). Arguing as in Step (VI) of the previous section we can show that I n 2 → 0 as n → 0. To deal with I n 1 we first use Assumption 4(iii) given on page 16 and (9.12) to derive that P n -a.s. F n (·,û n ) − F (·, u * ) r L p * (0,T ;X) → 0, as n → ∞. Since, by Eqs. 9.10, 9.16 and 9.17, we can apply the Lebesgue DCT and deduce that I n 1 → 0 as n → ∞. Therefore, as n → ∞ as n → ∞. These two facts along with Eq. 9.15 implies thatP-a.s. and for a.e. t ∈ [0, T ] u * (t) = K(x, u * , η * )(t).

It follows now from Step(IV)(ii), Step(V)(ii) and
Step(VI) that the system (ˆ ,F ,F,P, η * , u * ) is an X-valued martingale solution to problem (3.1) with càdlàg paths in B 0 := D(A ρ −1 ) for any ρ ∈ (0, ρ). Since −A is the infinitesimal generator of a contraction type C 0 -semigroup on D(A ρ− 1 p ) and u * ∈ Lq (0, T ; X) almost surely, then we easily infer from Lemma 7.3-(iii) that the paths of u * are càdlàg in D(A ρ− 1 p ). Similar calculations as done in Steps (I) and (II), see also Remark 9.2, yield that for anyq ∈ (q, q max ) and r ∈ (1, p) This completes the proof of Theorem 3.2.
for discussion related to the dual predictable projection of a Poisson random measure and to Szymon Peszat for discussion related to an example from his paper [64].
We also would like thank the anonymous referees for their insightful comments and help to clarify issues from previous version of the paper; in particular for their help in clarifying the construction of stochastic integral with respect to Poisson random measure (PRM) and progressively measurable integrands.
Last but not least, the authors would like to thank Carl Chalk, Pani Fernando, Ela Motyl, Markus Riedle, Akash Panda and Nimit Rana for a careful reading of the manuscript. Earlier versions of this paper can be found on arXiv:1010.5933.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

A.1 Space-time Poissonian White Noise
Analogously to the space-time Gaussian white noise one can construct a space-time Lévy white noise or space-time Poissonian white noise. But before doing this, let us recall the definition of a Gaussian white noise, see for e.g. Dalang [22].  As in the case of space-time Gaussian white noise we also introduce the following definition.
Definition A. 5 Let us assume that ( , F, P) is a complete probability space and ν be a Lévy measure on R. In what follows, for any f ∈ L p (R d ) we denote by f δ a the value of (f ) at a. Observe that f δ a , φ = f (a)φ(a), φ ∈ S(R d ) so that f δ a = f (a)δ a . Hence, in order to prove (B.1) it is sufficient to prove it for f = 1, i.e., Let us recall the definition of the Besov spaces as given in [69, Definition 2, pp. 7-8]. First we choose a function ψ ∈ S(R d ) such that 0 ≤ ψ(x) ≤ 1, x ∈ R d and We will use the definition of the Fourier transform F = F +1 and its inverse F −1 as in [69, p. 6]. In particular, with ·, · being the scalar product in R d , we put With the choice of φ = {φ j } ∞ j =0 as above and F and F −1 being the Fourier and the inverse Fourier transformations (acting on the space S (R d ) of Schwartz distributions) we have the following definition. If q = ∞ we put We denote by B s p,q (R d ) the space of all f ∈ S (R d ) for which |f | B s p,q is finite.
Proof of Theorem C.1. The proof is a modification of the proof of [24, Chapter 2, Theorem 2.4]. For simplicity, let us put PMU n := Law(χ n ), PMU 1 n := Law(π 1 • χ n ), n ∈ N, and PMU ∞ := lim n→∞ L(χ n ). We will generate families of partitions of U 1 and U 2 . To start with let (x i ) i∈N and (y i ) i∈N be dense subsets in U 1 and U 2 , respectively, and let (r i ) i∈N be a sequence of natural numbers converging to zero. Some additional condition on the sequence will be given below. For simplicity, we enumerate for any k ∈ N these families and call them (O k i ) i∈N , and (C k j ) j ∈N . Let¯ := [0, 1) × [0, 1) and Leb be the Lebesgue measure on [0, 1) × [0, 1). In the first step, we will construct a family of partition consisting of rectangles in¯ .
Definition C.2 Suppose that μ is a Borel probability measure on U = U 1 × U 2 and μ 1 is the marginal of μ on U 1 , i.e., μ 1 (O) := μ(O × U 2 ), O ∈ B(U 1 ). Assume that (O i ) i∈N and (C i ) i∈N are partitions of U 1 and U 2 , respectively. Define the following partition of the square [0, 1) × [0, 1). For i, j ∈ N we put

Remark C.3
Obviously, if μ 1 (O i ) = 0 for some i ∈ N, then I ij = ∅ for all j ∈ N.
Next for fixed l ∈ N and n ∈ N, we will define a partition I l,n ij i,j ∈N of¯ = [0, 1) × [0, 1) corresponding to the partitions (O l i ) i∈N and (C l i ) i∈N of the spaces U 1 and U 2 , respectively.
We denote by μ(O|C) the conditional probability of O under the condition C. Then, we have for n ∈ N where a n,k = PMU n (U 1 × C 1 1 | O 1 k × U 2 ) and b n = PMU n (U 1 × C 1 1 | O 1 1 × U 2 ) + PMU n (U 1 × C 1 2 | O 1 1 × U 2 ). More generally, for k ∈ N Finally, for k, l, r ≥ 2 Let us observe that for fixed l ∈ N, the rectangles {I l k,r : k, r ∈ N} are pairwise disjoint and the family {I l k,r : k, r ∈ N} is a covering of¯ . Therefore, we conclude that for any n ∈ N ∪ {∞} we have PMU n (U 1 × U 2 ) = 1 and m∈N PMU n (U 1 × C l m | O l k ) = 1. Consequently, it follows that for fixed l, n ∈ N the family of sets {I k,r : k, r ∈ N} is a covering of [0, 1) × [0, 1) and consists of disjoint sets. The next step is to construct the random variablesχ n :¯ → U 1 × U 2 , such that Law(χ n ) = Law(χ n ). We assume that r m is chosen in such a way, that the measure of the boundaries of the covering (O j ) j ∈N and (C j ) j ∈N are zero. In each non-empty sets int(O m j ) and int(C m j ) we choose points x m j and y m j , respectively, from the dense subsets (x i ) i∈N and (y i ) i∈N and define the following random variables. First, we put for m ∈ N Z 1 n,m (ω) = x m k ifω ∈ I m,n k,r , Z 2 n,m (ω) = y m r ifω ∈ I m,n k,r , n ∈ N ∪ {∞}, and then, for n ∈ N ∪ {∞}χ Due to the construction of the partition, the limits above exist. To be precise, for any n ∈ N ∪ {∞} andω ∈¯ , we have and therefore (Z n,m (ω)) m≥1 is a Cauchy-sequence for allω ∈¯ = [0, 1) × [0, 1). Hence, Z i n (ω), i, n, is well defined. Furthermore, χ n is measurable, since Z i n,m are simple functions, hence measurable. Therefore, Z i n (ω), i, n, is a random variable. Finally, we have to proof that the random variablesχ andχ n := (χ n 1 ,χ n 2 ) have the following properties: (i) Law(χ n ) = Law(χ n ), ∀n ∈ N, (ii) χ n → χ a.s. in U 1 × U 2 , (iii) π 1 • χ n (ω) = π 1 • χ * (ω).
Proof of (ii) We will first prove that there exists a random variable χ = (χ 1 , χ 2 ) such that χ n 1 → χ 1 andχ n 2 → χ 2 Leb-a.s. for n → ∞. For this purpose it is enough to show that the sequences (χ n 1 ) n∈N and (χ n 2 ) n∈N are Leb-a.s. Cauchy sequences. From the triangle inequality we infer that for all n, m, j ∈ N, i = 1, 2 χ n i −χ i m ≤ χ i n − Z i n,j + Z i n,j − Z i m,j + Z i m,j −χ i m .

Moreover, since
Leb where q = p 1−pα . Since W α,p (I 0 ) ⊂ L r (I 0 ) for any r ∈ [1, p 1−pα ] we infer from the last inequality that there exists C > 0 such that