Yamada-Watanabe Theorem for Stochastic Evolution Equation Driven by Poisson Random Measure

The main purpose of this paper is to establish the YamadaWatanabe theory of uniqueness and existence of solutions of stochastic evolution equation driven by pure Poisson random measure in the variational approach. The classical paper [1] has initiated a comprehensive study of relations between different types of uniqueness and existence (e.g., strong solutions, weak solutions, pathwise uniqueness, uniqueness, and joint uniqueness in law) arising in the study of SDEs (see, e.g., [2–4]) and the study is still alive today. New papers are published (see, e.g., [2, 3, 5–7]). In this paper we are concerned with the similar question for stochastic evolution equation driven by Poisson random measure by using the method of Yamada and Watanabe. Yamada andWatanabe’s initial work [1] proved that weak existence and pathwise uniqueness imply strong existence and weak uniqueness. For n-dimensional case, see [8, 9]. For infinite dimensional stochastic differential equation, Ondreját [6] proved similar result for stochastic evolution equation in Banach space driven by cylindrical Wiener process, where the solutions are understood in the mild sense. Lately, Röckner et al. [7] proved similar result for stochastic evolution equation in Banach space driven by cylindrical Wiener process under the variational framework. On the other hand, Kurtz [2, 3] obtained a pleasant version of Yamada-Watanabe and Engelbert theorem in an abstract form, which covered most of the work mentioned above. However, we will consider the following concrete stochastic evolution equation by using a different method. In this paper, we will consider the following stochastic evolution equation driven by pure Poisson random measure under the variational framework:


Introduction
The main purpose of this paper is to establish the Yamada-Watanabe theory of uniqueness and existence of solutions of stochastic evolution equation driven by pure Poisson random measure in the variational approach.The classical paper [1] has initiated a comprehensive study of relations between different types of uniqueness and existence (e.g., strong solutions, weak solutions, pathwise uniqueness, uniqueness, and joint uniqueness in law) arising in the study of SDEs (see, e.g., [2][3][4]) and the study is still alive today.New papers are published (see, e.g., [2,3,[5][6][7]).In this paper we are concerned with the similar question for stochastic evolution equation driven by Poisson random measure by using the method of Yamada and Watanabe.
Yamada and Watanabe's initial work [1] proved that weak existence and pathwise uniqueness imply strong existence and weak uniqueness.For -dimensional case, see [8,9].For infinite dimensional stochastic differential equation, Ondreját [6] proved similar result for stochastic evolution equation in Banach space driven by cylindrical Wiener process, where the solutions are understood in the mild sense.Lately, Röckner et al. [7] proved similar result for stochastic evolution equation in Banach space driven by cylindrical Wiener process under the variational framework.On the other hand, Kurtz [2,3] obtained a pleasant version of Yamada-Watanabe and Engelbert theorem in an abstract form, which covered most of the work mentioned above.However, we will consider the following concrete stochastic evolution equation by using a different method.
In this paper, we will consider the following stochastic evolution equation driven by pure Poisson random measure under the variational framework: This type of equations can be applied to many SPDEs, for example, stochastic Burgers equation, stochastic porous media equation, and stochastic Navier-Stokes equation (see, e.g., [9][10][11][12][13]).We will introduce the above equation precisely in Section 2. Our aim is to obtain this jump-case Yamada-Watanabe theorem; that is, weak existence and strong uniqueness (which will be stated in Section 2) imply strong existence and weak uniqueness and vice versa.We note that there are some differences between the jump-case case and the Brownian motion case.It is well known that a Brownian motion can be treated as a canonical map on ([0, ∞); R  ) or ([0, ∞); ) (for some Hilbert space ), while for jump-case we have to work on the configuration space N (see Section 2) for Poisson random measure.
The structure of the present paper is as follows.In Section 2 we introduce the framework and definitions of strong solution and weak solution.In Section 3 we give and prove our main results.

Framework and Definitions
Let  be a separable Hilbert space with inner product ⟨⋅, ⋅⟩  and norm ‖ ⋅ ‖  .Let  and  be separable Banach space with norms ‖ ⋅ ‖  and ‖ ⋅ ‖  , such that continuously and densely.
Let  be a -finite measure on (X, B(X)).We recall that a Poisson random measure  on X with intensity  is a measurable mapping from a probability space (Ω, F, ) to (P Z + (X), A) such that the following two conditions hold: (i) for every  ∈ B(X), the random variable () has the Poisson distribution with the mean (); (ii) for any disjoint  1 ,  2 , . . .,   ∈ B(X), the random variables ( 1 ), ( 2 ), . . ., (  ) are independent.
Let  be the distribution of ; then (P Z + (X), A, ) is a probability space, and the canonical map is a Poisson random measure with intensity  on this probability space.Let N := P Z + (X).Note that N is a Polish space (cf.[15]).In this paper, let X := [0, ∞) × , where  is an arbitrary locally compact Hausdorff topological space with countable base, and  = ](), where ] is a -finite measure on .We denote by B  (N) the -algebra generated by the mappings: N ∋  → ((0, ] × ) ∈ Z + , for all 0 ⩽  ⩽ , for all  ∈ B().
Definition 1. Assume that (Ω, F, , (F  )) is a filtered probability space and  is a Poisson random measure defined as above.Then  is called a (ii) as a stochastic equation on  one has Definition 4. One says that weak uniqueness holds for (7) if whenever (, ) and (  ,   ) are two weak solutions with stochastic bases (Ω, F, , (F  )) and (Ω  , F  ,   , (F   )) such that then Definition 5.One says that pathwise uniqueness holds for (7) if whenever (, ) and (  , ) are two weak solutions on the same stochatic bases (Ω, F, , (F  )) such that (0) =   (0) -a.s.; then -a.s.: In order to define strong solutions one needs to introduce the following class Γ of maps.Let Γ denote the set of all maps  :  × N → D such that for every probability measure  7) on (Ω, F, , (F  )) is said to be a strong solution if there exists  ∈ Γ with respect to  ∘ (0) −1 such that, for  ∈ ,  → (, ) is B  (N) /B  (D)-measurable for every  ∈ [0, ∞) and where B  (N) denote the completion with respect to  in B(N).Definition 7. Equation ( 7) is said to have a unique strong solution, if there exists  ∈ Γ satisfying the condition in the above definition such that the following conditions hold.

The Main Result and Its Proof
Let us now state the main result.
Theorem 8. Let  and  be as above.Then (7) has a unique strong solution if and only if both of the following properties hold.
(i) For every probability measure  on (, B()) there exists a weak solution (, ) of ( 7) such that  is the distribution of (0).
Proof.Suppose that (7) has a unique strong solution.Then (ii) obviously holds.To show (i) we only have to take the probability space (N, B(N), ) and consider ( × N, B() ⊗ B(N) ⊗ ,  ⊗ ) with filtration where N denotes all  ⊗ -zero sets in B() ⊗ B(N) ⊗ .Let  :  × N →  and :  × N → N be the canonical projections.Then ( :=  ∘ −1 (, ), ) is the desired weak solution in (i).Now let us suppose that (i) and (ii) hold.We are going to show that there exists a unique strong solution for (7).
Let (, ) with stochastic basis (Ω, F, , (F  )) be a weak solution to (7)  (18) We have the following lemma.( Proof.Let Π : ×D×N → ×N be the canonical projection.Since (0) is F 0 -measurable, hence -independent of , it follows that We recall that D is a Polish space and, by the existence result on regular conditional distributions the family   ((, ), ),  ∈ ,  ∈ N, exist and satisfy (i) and (ii).
For  ∈  define a measure   on by Define the stochastic basis where and define the maps Then, it is easy to see that Lemma 10.There exists  ∈ B() with () = 0 such that, for all  ∉ , one has that Π 3 is an ( F  )-Poisson random measure on ( Ω, F ,   ).