Uniqueness of the nonlinear Schrödinger equation driven by jump processes

Transkript

1 Nonlinear Differ. Equ. Appl. (219) 26:22 c 219 pringer Nature witzerland AG /19/31-31 published online June 6, Nonlinear Differential Equations and Applications NoDEA Uniqueness of the nonlinear chrödinger equation driven by jump processes Anne de Bouard, Erika Hausenblas and Martin Ondreját Abstract. In a recent paper by the first two authors, existence of martingale solutions to a stochastic nonlinear chrödinger equation driven by alévy noise was proved. In this paper, we prove pathwise uniqueness, uniqueness in law and existence of strong solutions to this problem using an abstract uniqueness result of Kurtz. Mathematics ubject Classification. Primary 6H15; econdary 6G57. Keywords. Uniqueness results, Yamada Watanabe Kurtz theorem, tochastic integral of jump type, tochastic partial differential equations, Poisson random measures, Lévy processes, chrödinger equation. 1. Introduction This paper is a natural continuation of [11], where the first two authors proved the existence of global solutions to a stochastic non-linear chrödinger equation driven by a time homogeneous Poisson random measure. The existence of solutions was proved in the weak probabilistic sense, i.e., on an undetermined stochastic basis. The present paper aims to give sufficient conditions under which these solutions are also strong and unique. We proceed first by proving pathwise uniqueness of the solutions, and then we apply a result of Yamada Watanabe Kurtz to show that these solutions are strong and unique in law. The Yamada Watanabe theory has been well developed for stochastic equations driven by Wiener processes, see e.g. [8,12,14,21,23,25,3] or[1] for forward-backward stochastic differential equations. There are also analogous results for equations driven by Poisson random measures. For instance, in [2], the authors develop the Yamada Watanabe theory for stochastic differential equations driven by both a Wiener process and a Poisson random The second author was supported by the FWF-Project P17273-N12. The research of the third author on this work was supported by the GAČR Grant No. GA

2 22 Page 2 of 31 A. de Bouard, E. Hausenblas, and M. Ondreját NoDEA measure (where the latter is defined on a locally compact space) via the original method of Yamada and Watanabe [3]. In [31], the Yamada Watanabe theory is presented for variational solutions of partial differential equations driven by a Poisson random measure on a locally compact space also by the method of Yamada and Watanabe [3]. Unfortunately, none of these results applies to our problem, mainly because the noise does not live in a locally compact space. Let us remind the reader that the original proof of Yamada and Watanabe is based on an application of a theorem on the existence of a regular version of a conditional probability. Their idea has proved to be so strong and robust to be applicable not only to stochastic differential equations but also to stochastic partial differential equations driven by various noises. In 27, Kurtz [17] presented an abstract Yamada Watanabe theory aiming not only at stochastic equations but also at many other stochastic problems of different nature. In that paper, Kurtz was the first one to abandon the original idea of the proof of Yamada and Watanabe (regular version of a conditional probability) and used the universal korokhod representation theorem. This approach made it possible to raise the Yamada Watanabe theory to an abstract level (see also [18]) where details of particular problems, to which it is applicable, play no role. On the other hand, this abstract approach has one disadvantage for applications that everyone who wishes to apply the result must translate his particular problem to the language of [17] which itself is not straightforward. In this paper, we recourse to [17] due to its generality. We consider mild solutions to stochastic partial differential equations in Banach spaces driven by time homogeneous Poisson random measures on a Polish space (which is not in general locally compact, so we cannot apply the results [2,31]), we translate this problem to the language of [17] and prove the standard existence of unique strong solutions. This result is then applied to the stochastic nonlinear chrödinger equation driven by a time homogeneous Poisson random measure. To be more precise, let A = Δ be the Laplace operator with D(A) = {u L 2 (R d ):Δu L 2 (R d )}. We are interested in the solution of the following equation idu(t, x) Δu(t, x) dt + λ u(t, x) α 1 u(t, x) dt = u(t, x) g(z(x)) η(dz, dt)+ u(t, x) h(z(x)) ν(dz) dt, t [,T], (1.1) u() = u, where η denotes the Poisson random measure corresponding to L and η the compensated Poisson random measure, ν the intensity of the Poisson random measure. This equation can be rewritten in terms of a Lévy process having characteristic measure ν. For more details on the connection of Lévy processes and Poisson random measure we refer to section 2.3 in [7] and the references therein. We would like to remark that Poisson random measures are more general than Lévy processes. If the stochastic perturbation is a Wiener process, the equation is well treated, and the existence and uniqueness of the solution are known. For more

3 NoDEA Uniqueness of the nonlinear chrödinger equation Page 3 of information see [9,1]. In case the stochastic perturbation is replaced by a Lévy process with infinite activity, de Bouard and Hausenblas could only show in [11] the existence of a solution, without uniqueness. Here in this work, we are interested in conditions under which a unique solution exists. ince we will use it later on, we will introduce some notations. Notation 1.1. R denotes the real numbers, R + := {x R : x > } and R + := R+ {}. ByN we denote the set of natural numbers (including ) and by N we denote the set N { }. Notation 1.2. If (F t ) t [,T ] is a filtration and θ a measure then we denote by Ft θ the augmentation of F t by the θ-null sets in F. Notation 1.3. The set of all finite non-negative measures on a Polish space (, ) will be denoted by M + () andp 1 () will stand for probability measures on. If a family of sets { n : n N} satisfy n then M N({ n }) denotes the family of all N-valued measures θ on such that θ( n ) < for every n N. ByM N({ n }) we denote the σ-field on M N({ n }) generated by the functions i B : M N({ n }) μ μ(b) N, B. 2. Time homogeneous Poisson random measures ince the definition of time homogeneous Poisson random measure is introduced in many, not always equivalent ways, we give here our definition. Definition 2.1. (ee [13], Def. I.8.1) Let (, ) be a Polish space, ν a σ-finite measure on (, ), { n }such that n and ν( n ) < for every n N. A time homogenous Poisson random measure η over a filtered probability space (Ω, F, F, P), where F =(F t ) t [,T ], is a measurable function η :(Ω, F) (M N({ n (,T]}), M N({ n (,T]})), such that (i) for each B B(R + ) with Eη(B) < η(b) :=i B η :Ω N is a Poisson random variable with parameter Eη(B), otherwise η(b) = a.s. (ii) η is independently scattered, i.e. if the sets B j B(R + ), j =1,...,n, are disjoint, then the random variables η(b j ), j =1,...,n, are mutually independent; (iii) for each U,the N-valued process (N(t, U)) t [,T ] defined by N(t, U) :=η(u (,t]), t [,T] is F-adapted and its increments are stationary and independent of the past, i.e. if t>s, then N(t, U) N(s, U) =η(u (s, t]) is independent of F s. Remark 2.2. In the framework of Definition 2.1 the assignment ν : A E [ η(a (, 1)) ] defines a uniquely determined measure, called in the following intensity measure.

4 22 Page 4 of 31 A. de Bouard, E. Hausenblas, and M. Ondreját NoDEA Besides, the term Poisson random measure is sometimes defined in another way, i.e., by starting with the intensity measure and defining the Poisson random measure with a given intensity measure. However, we prove in the next lemma the equivalence between both definitions. Lemma 2.3. A measurable mapping η : Ω M N({ n (,T]}) is a time homogeneous Poisson random measure with intensity ν iff (a) for any U with ν(u) <, the random variable N(t, U) is Poisson distributed with parameter tν(u), otherwise P (N(t, U) = ) =1; (b) for any n and disjoint sets U 1,U 2,...,U n,andanyt [,T], the random variables N(t, U 1 ), N(t, U 2 ),..., N(t, U n ) are mutually independent; (c) the M N({ n })-valued process (N(t, )) t (,T ] is adapted to F; (d) for any t [,T], U, ν(u) <, andanyr, s t, the random variables N(r, U) N(s, U) are independent of F t. Proof. To see the equivalence of (i) and (a), first observe, that if is a separable metric space, the Borel space B( (,T]) of the cartesian product R + is the product of the Borel spaces B() andb(r + ), see [22, p. 6, Theorem 1.1]. This implies the equivalence from (i) to (a). To show the equivalence of (ii) and (b), one has, in addition, to take into account that the Borel σ-algebra can be generated by intervals of the form {(,t]:t (,T]}. The equivalence of (iii), and (c) and (d) follows by the definition of N(t, U). Usually, one starts with specifying the measurable space (, ) and the intensity measure ν on (, ). Given this, then there exists a Poisson random measure on (, ) having the intensity measure ν. In order to define a stochastic integral with respect to the Poisson random measure, has to be related to a topological vector space, and the measure ν has either to be finite or has to be a Lévy measure. The definition of Lévy measures in Banach spaces can be found in [2, Chapter 5.4]. Here, we give a characterization of Lévy measures, which is sufficient for our purpose. Remark 2.4. Let be a separable Banach space, and its Borel σ-algebra. If the intensity measure ν : Rsatisfies the integrability condition sup a a 1 1 z,a 2 ν(dz) <. then ν is a Lévy measure (see [2, Proposition 5.4.1, p. 7]). For some Banach spaces, one can characterize the Lévy measures in a more precise way. Therefore, let us introduce the following definition. Let {ε k : k N} be a sequence of independent, identically distributed random variables with P (ε 1 =1)=P (ε 1 = 1) = 1 2. Then a Banach space with norm is of R- type p, (Rademacher type p), where 1 p 2, if for any sequence {x j : j N} belonging to l p (E), we have (compare [2, p. 4]) P ε j x j < =1. j=1

5 NoDEA Uniqueness of the nonlinear chrödinger equation Page 5 of The Minkowski inequality implies that each Banach space is of R-type 1. Remark 2.5. Let be a Polish space, the Borel σ-algebra over (in the sequel we call (, ) just a Polish space), let { n }satisfy n, letν be a σ-finite measure with ν( n ) < for every n N and fix p [1, 2]. We assume that E is a separable Banach space of R-type p and that ξ :(, ) (E,B(E)) is a measurable mapping. In addition, we assume that the intensity measure ν : R + satisfies the integrability condition 1 ξ(z) p E ν(dz) <, and ν({}) =. (2.1) Then, the measure ν E induced by ξ on E is a Lévy measure (and ν E ({}) :=) (compare [2, p. 75]). In addition, if η is a Poisson random measure with intensity ν over a filtered probability space (Ω, F, F, P), the process L :[,T] t ξ(z )(η ν λ)(dz, ds) is a Lévy process over (Ω, F, F, P). Here λ denotes the Lebesgue measure on R. Hence, from now on we will assume during the whole paper that the following convention is valid. Convention 2.1. We convene that (, ) is a Polish space, ν a σ-finite measure on (, ) and n such that n and ν( n ) < for every n N. Let us consider a filtered probability space (Ω, F, F, P), where F = {F t } t [,T ] denotes a filtration. A process ξ :[,T] Ω X is progressively measurable, or simply, progressive, if its restriction to Ω [,t]isf t B([,t])- measurable for any t. The predictable random field P on Ω R + is the σ-field generated by all continuous F-adapted processes (see e.g., Kallenberg [15, Chapter 25, p. 491]). A real valued stochastic process {x(t) :t [,T]}, defined on a filtered probability space (Ω, F, F, P) is called predictable, if the mapping x :Ω (,T] R is P/B(R)-measurable. A random measure γ on B((,T]) over (Ω; F, F, P) is called predictable, iff for each U,theR-valued process (,T] t γ(u (,t]) is predictable. Definition 2.6. Assume that (, ) is a measurable space and ν is a nonnegative σ-finite measure on (, ). Assume that η is a time homogeneous Poisson random measure with intensity measure ν on (, ) over(ω, F, F, P). The compensator of η is the unique predictable random measure, denoted by γ, on B((,T]) over (Ω, F, F, P), such that for each T < and A with Eη(A (,T]) <, ther-valued processes {Ñ(t, A)} t (,T ] defined by Ñ(t, A) :=η(a (,t]) γ(a (,t]), <t T, is a martingale on (Ω, F, F, P).

6 22 Page 6 of 31 A. de Bouard, E. Hausenblas, and M. Ondreját NoDEA Remark 2.7. Assume that η is a time homogeneous Poisson random measure with intensity ν on (, ) over(ω, F, F, P). It turns out that the compensator γ of η is uniquely determined and moreover γ : B(R + ) (A, I) ν(a) λ(i). The difference between a time homogeneous Poisson random measure η and its compensator γ, i.e. η = η γ, is called a compensated Poisson random measure. Let (, ) be a measurable space and let η be a time homogenous Poisson random measure on with intensity measure ν being a positive σ-finite measure over A satisfying Convention 2.1. We will denote by η the compensated Poisson random measure defined by η := η γ. Lemma 2.8. Let ν be a non-negative σ-finite measure on satisfying Convention 2.1. Then the following holds (i) there exists a probability space A =(Ω, F, P) and a time homogenous Poisson random measure η :Ω M N({ n (,T]}) with the intensity measure ν; (ii) Denote by Θ ν the law of η on M N({ n (,T]}). Ifη is a time homogenous Poisson random measure defined possibly on different stochastic base A =(Ω, F, P ) and ν is the intensity measure for η then Θ ν is the law of η on M N({ n (,T]}). Proof. Part (i) is given by Theorem 8.1 [13, p. 42]. It remains to show (ii). ince ν is σ-finite, there exists an increasing family { n : n N} with n+1 n, n, andν( n ) <. To show that η and η have the same law on M N( (,T]), we have to show that for all f : (,T] R, bounded and continuous, the random variable η(f) := T n f(s, t) η(ds, dt) andη (f) := T n f(s, t) η (ds, dz) have the same law, see [22, Theorem 5.8, p. 38]. ince R + is a Polish space, the σ algebra generated by the family of bounded continuous functions coincides with the Borel-σ-algebra, see [26, Proposition 1.4, p.5]. Therefore, it is sufficient to show for all n N, U B( n )and I B((,T]), that the random variables η(u I) andη (U I) havethe same law. Let Θ ν be the law of η and let us assume ν(u),λ(i) <. Let k N. Then, by the definition of the Poisson random measure and its intensity measure ν we know that λ(i)ν(u) (ν(u) λ(i))k Θ ν (η(u I]) = k) =e k! =Θ ν(η (U I) =k). If ν(u) = or λ(i) =, then Θ ν (η(u I) = ) =1=Θ ν (η(u I) = ). Now, one can define the stochastic integral with respect to the Poisson random measure for progressively measurable integrands, introduced, e.g., in [5] in M-type p Banach spaces.

7 NoDEA Uniqueness of the nonlinear chrödinger equation Page 7 of Definition 2.9. Let <p 2. A Banach space E is of martingale type p iff there exists a constant C> such that for all E-valued finite martingale {M n } N n= the following inequality holds sup n N where as usually, we put M 1 =. N E M n p E C E M n M n 1 p E, (2.2) Examples of M-type p Banach spaces are, e.g., L q (O) spaces, where O is a bounded domain. L q (O) isofm-type p for any p q (see e.g. [29, Chapter 2, Example 2.2]). If a Banach space E is of M-type p and A is the generator of an analytic semigroup on E, then the complex interpolation spaces between D(A) ande are of M-type p. imilar facts also hold for real interpolation spaces, but not in this generality, for more details we refer to Appendix A of [4]. In particular, in [5] it is proven that for any Banach space E of M-type p there exists a unique continuous linear operator I which associates to each progressively measurable process ξ : R + Ω L p (, ν; E) with P-a.s. T ξ(r, x) p E ν(dx)dr < (2.3) n= for every T>, an adapted E-valued càdlàg process I ξ, η (t) := ξ(r, x) η(dr, dx), t such that if a process ξ satisfying the above condition (2.3) is a random step process with representation n ξ(r, x) = 1 (tj 1,t j](r) ξ j (x), x, r [,T], (2.4) j=1 where {t =<t 1 <... < t n < } is a finite partition of [, ) and for all j {1,...,n}, ξ j is an E-valued F tj 1 -measurable p-summable simple random variable, then n I ξ, η (t) = ξ j (x) η ((t j 1 t, t j t],dx), t [,T]. (2.5) j=1 This definition can be extended to all progressively measurable mappings ξ : Ω [,T] E with P-a.s. T min(1, ξ(r, z) p E )ν(dz) dr <. ome information on the different settings is given in [24]. Here, in this work we will only use p = 2, however, for the sake of completeness, we assume p [1, 2]. Besides, we would like to point out in the following Proposition, that we do not need to suppose that the filtration of the given probability space is right continuous. In particular, given a Poisson random measure η over a

8 22 Page 8 of 31 A. de Bouard, E. Hausenblas, and M. Ondreját NoDEA filtered probability space (Ω, P, F, F), F =(F t ) t [,T ], with an arbitrary filtration, a progressively measurable L 2 (, ν)-valued process ξ, one can pass to the right continuous augmentation of the filtration without loosing the necessary properties. The following remark can be easily shown. Remark 2.1. Let us assume that η is also a Poisson random measure for the augmented right continuous filtration F t := h> Ft+h P. Then, we can construct the stochastic integral on (Ω, F, (F t ), P), and the stochastic integral on (Ω, F, ( F t ), P). In particular, let I 1 =(F t ) ξ(r, x) η(dr, dx), resp. I 2 =( F t ) ξ(r, x) η(dr, dx), where in the integral on the left hand, we took for the predictable sequence of simple functions (ξ n ) converging to ξ the underlying filtration (F t ) t and in the integral on the right hand we took for predictable sequence of simple functions (ξ n ) converging to ξ the underlying filtration ( F t ) t. Then, I 1 = I Pathwise uniqueness of the stochastic chrödinger equation We are interested in the uniqueness of the stochastic chrödinger equation driven by a Lévy noise. The nonlinear chrödinger equation is an example of a universal nonlinear model that describes many physical nonlinear systems. The equation can be applied to hydrodynamics, nonlinear optics, nonlinear acoustics, quantum condensates, heat pulses in solids and various other nonlinear instability phenomena. The chrödinger equation also arises in the context of water waves. In 1968 V.E. Zakharov derived the Nonlinear chrödinger equation for the two-dimensional water wave problem in the absence of surface tension, that is, for the evolution of gravity-driven surface water waves. More recently, Villarroel, et all [27, 28] considered the nonlinear chrödinger equation with randomly distributed, but isolated jumps. Dealing with jumps one may model sudden changes in the field that occur randomly. In order to model abrupt changes in the medium, as it can, e.g., be the case for the propagation of light in optical fibers, or in other media, one can use Lévy noise. In [11] the first and second authors investigated the existence of a solution if the Lévy measure has infinite activity. However, no uniqueness was proven. From now on, will be a Borel subset of a separable Banach function space continuously embedded in the obolev space W (R 1 d ). As mentioned in the introduction, the equation we are considering is given by idu(t, x) Δu(t, x) dt + λ u(t, x) α 1 u(t, x) dt = u(t, x) g(z(x)) η(dz, dt)+ u(t, x) h(z(x)) γ(dz, dt), t [,T], (3.1) u() = u. with λ. Here, g : R C and h : R C are two functions satisfying : (i) There exist some constants C g and C h such that g(ξ) C g ξ, g (ξ) C g ξ and h(ξ) C h ξ, h (ξ) C h ξ.

9 NoDEA Uniqueness of the nonlinear chrödinger equation Page 9 of Note that this implies : (ii) g() = and h() =, (iii) I(h(ξ)) ξ 2 and I(g(ξ)) ξ 2. Here, I denotes the imaginary part of a number. The integrability condition (2.1) for the intensity measure ν and condition (i) gives the necessary conditions for the corresponding Nemytskii operator induced by g. Let us denote by (T (t)) t the group of isometries generated by the operator ia. In particular, for any t R and u L 2 (R d ) let us denote the solution of the following Cauchy problem { i u(t) =Au(t), u() = u, by T (t)u. Observe, (T (t)) t forms a unitary group on L 2 (R d )andh γ 2 (Rd ) for any γ R. In the framework of evolution equations, one considers the mild solution of Eq. (3.1), which is given by the following integral equation for t [,T] t u(t) =T (t)u + iλ T (t s)( u(s) α 1 u(s)) ds i T (t s)[u(s) G(z)] η(dz, ds) i T (t s)[u(s)h(z)] ν(dz) ds. Here, we used the Nemytskii operators corresponding to g and h. In particular, the mappings G : L 2 (R d ) L 2 (R d )andh : L 2 (R d ) L 2 (R d ) denote the Nemytskii operators associated to the functions g and h defined by (G(y))(x) :=g(y(x)), and (H(y))(x) :=h(y(x)), y, x R d. Definition 3.1. Let T >. We call u an L 2 (R d )-valued mild solution to Eq. (3.1) on the time interval [,T], iff u is an adapted cádlág process in L 2 (R d ), the terms t T (t s)[ u(s) α 1 u(s)] L 2 ds + T (t s)[u(s) H(y)] L 2 ν(dy) ds + T (t s)[u(s) G(y)] L 2 ν(dy) ds, {y : T (t s)u(s) G(y) L 2 1} + T (t s)[u(s) G(y)] 2 L2 ν(dy) ds {y : T (t s)u(s) G(y) L 2 <1} < P-a.s for every t [,T]

10 22 Page 1 of 31 A. de Bouard, E. Hausenblas, and M. Ondreját NoDEA and for any t [,T], the process u solves P-a.s. the integral equation t u(t) =T (t)u + iλ T (t s)[ u(s) α 1 u(s)] ds i T (t s)[u(s)g(y)] η(dy, ds) i T (t s)[u(s)h(y)] ν(dy) ds. (3.2) ince in the proof of the existence of solution compactness arguments are used, the underlying probability space gets lost. Hence, a concept of probabilistic weak solutions has to be introduced, which is done in the following definition. Definition 3.2. Let u L 2 (R d ). A martingale solution on L 2 (R d )totheproblem (3.1) is a system (Ω, F, P, F,η,u) (3.3) such that (i) (Ω, F, F, P) is a filtered probability space with filtration F =(F t ) t [,T ], (ii) η is a time homogeneous Poisson random measure on (, ) over (Ω, F, F, P) with intensity measure ν satisfying Convention 2.1, (iii) u is an L 2 (R d )-valued mild solution to Eq. (3.1). Let us remind that is a function space continuously embedded in the obolev space W (R 1 d ). Besides, we assume that the intensity measure satisfies the following integrability conditions: (i) C (ν) := z 2 L ν(dz) < ; (ii) C 1 (ν) := z 2 W < ; ν(dz) 1 (iii) C 2 (ν) := x 2 z(x) 2 dx ν(dz) < ; R d (iv) C 3 (ν) := z 4 L ν(dz) <. Remark 3.3. One can see from the proof of Theorem 2.7 in [11] that the large jumps do not affect the uniqueness result and one can easily generalize our Theorem 3.4 to the case where one has large jumps without any bounded moments. Let { 1+4/(d 2) for d>2, 1 α< for d =1or2, In [11] we have shown that under the conditions stated above, there exists a martingale solution. For the sake of completeness, we state here the main result of the article. Before, since we will need it later on, let us introduce the mass defined by E(u) := u(x) 2 dx, (3.4) R d

11 NoDEA Uniqueness of the nonlinear chrödinger equation Page 11 of and the energy by H(u) := 1 u(x) 2 dx + λ u(x) α 1 u(x) u(x) dx. (3.5) 2 R α +1 d R d Theorem 3.4. Let η be a time homogenous Poisson random measure on with Lévy measure ν satisfying the integrability conditions (i), (ii), (iii), and (iv) given above. If u H2 1 (R d ), λ, and x 2 u (x) 2 dx <, R d then there exists a H2 1 (R d )-valued martingale solution to (3.1), which is cádlág in H γ 2 (Rd ) for any γ<1. In addition, there exists a constant C = C(T,C (ν),c 3 (ν),c g,c h ) such that E sup u(t) 2 L C (1 + E u 2 2 L 2) t T and for any T>there exists a constant C = C(T,C (ν),c 1 (ν),c g,c h ) > such that E sup H(u(t)) C (1 + EH(u )). t [,T ] The proof of Theorem 3.4 uses compactness arguments, hence, as mentioned before, the existence of a solution is shown, but no uniqueness of the solution. Here, we are interested in the uniqueness of the solution to Eq. (3.1). However, similar to the concept of solutions, there exist several concepts of uniqueness. Definition 3.5. The Eq. (3.1) ispathwise unique if, whenever (Ω, F, (F t ) t [,T ], P,η,u i ), i =1, 2 are solutions to (3.2) such that P (u 1 () = u 2 ()) = 1, then P (u 1 (t) =u 2 (t)) = 1 for every t T. Under certain conditions pathwise uniqueness of the stochastic chrödinger equation driven by Lévy noise can be shown. Theorem 3.6. Let us assume that g, h : R C are Lipschitz continuous. Let us assume that { 1 α< if d =1, or 2, 1 α< d (3.6) d 2 if d>2. Let be given a filtered probability space A =(Ω, F, F, P), with filtration F =(F t ) t [,T ], a Poisson random measure defined on A adapted to the filtration F, and two mild solutions u 1 and u 2 to Eq. (3.1) over A, on[,t] such that u 1 and u 2 are càdlàg in L 2 (R d ). If there exists some δ R with { d 2 δ> d 2(α 1), if d =1, 2, d 2 1 α 1, if d>2, (3.7) and the solutions u 1 and u 2 are belonging P-a.s. to D([,T]; H δ 2(R d )), then u 1 and u 2 are indistinguishable in L 2 (R d ).

12 22 Page 12 of 31 A. de Bouard, E. Hausenblas, and M. Ondreját NoDEA Remark 3.7. Condition (3.6) is needed to apply the trichartz estimate and obolev embedding theorems, in order to handle the nonlinearity. The restriction compared to usual conditions is due to the fact that we have to estimate the difference of solutions in L 2 (R d ) to tackle the stochastic terms. Proof of Theorem 3.6. Note that the solutions given by Theorem 1.2 of [11] satisfy the assumption of Theorem 3.6. Moreover, if u 1 and u 2 are as above, then they belong a.s. to L ([,T]; H2(R δ d )). In the first step we will introduce a family of stopping times {τ m : m N} and show that on the time interval [,τ m ] the solutions u 1 and u 2 are indistinguishable. In the second step, we will show that P (τ m <T) for m. From this follows that u 1 and u 2 are indistinguishable on the time interval [,T]. tep I Let us introduce the stopping times {τm 1 : m N} and {τm 2 : m N} given by τm i := sup{s >: u i (s) H δ 2 <m} T, i =1, 2. The aim is to show that u 1 and u 2 are indistinguishable on the time interval [,τ m ], with τ m = inf(τm,τ 1 m). 2 Fix m N. To get uniqueness on [,τ m ] we first stop the original solution processes at time τ m and extend the processes u 1 and u 2 by other processes to the whole interval [,T]. For this propose, let y 1 be a solution to y 1 (t) =T (t τ m )u 1 (τ m ) i T (t s)y 1 (s) G(z) η(dz, ds) τ m i T (t s)y 1 (s)h(z)γ(dz, ds), t τ m, (3.8) τ m and let y 2 be a solution to y 2 (t) =T (t τ m )u 2 (τ m ) i T (t s)y 2 (s) G(z) η(dz, ds) τ m i T (t s)y 2 (s)h(z)γ(dz, ds), t τ m. (3.9) τ m ince u 1 and u 2 are cádlág in L 2 (R d ), u 1 (τ m )andu 2 (τ m ) are well defined and belong P-a.s. to L 2 (R d ). ince, in addition, (T (t)) t R is a strongly continuous group on L 2 (R d ), the existence of unique solutions y 1 and y 2 to (3.8) and (3.9) in L 2 (R d ) can be shown by standard methods. Now, let us define two processes ū 1 and ū 2 which are equal to u 1 and u 2 on the time interval [,τ m ) and follow the linear chrödinger y 1 and y 2 afterwards. In particular, let equation { ū 1 (t) = u 1 (t) for t<τ m, y 1 (t) for τ m t T, and ū 2 (t) = { u 2 (t) for t<τ m, y 2 (t) for τ m t T.

13 NoDEA Uniqueness of the nonlinear chrödinger equation Page 13 of Note, that ū 1 and ū 2 solve the truncated equation corresponding to (3.2), that is u(t) =T (t)u + iλ ( ) t 1 [,τ m)(s) T (t s) u(s) α 1 u(s) ds i T (t s)u(s) G(z) η(dz, ds) i (3.1) t T (t s)u(s) H(z) γ(dz, ds). For u L 2 (R d )andξ N 2 (Ω; L 2 ([,T]; L 2 (R d ))) progressively measurable with respect to the filtration F, let us define the integral operator (Zξ)(t) :=T (t)u +(F τm ξ)(t)+(gξ)(t)+(hξ)(t), t [,T], (3.11) where u L 2 (R d ), the integral operator F τm with respect to the nonlinear term is defined by (F τm ξ)(t) =iλ G is defined by t (Gξ)(t) = i and H is defined by (Hξ)(t) := i T (t s) ( ξ(s) α 1 ξ(s) ) 1 [,τm)(s) ds, t t T (t s)ξ(s) G(z) η(dz, ds), T (t s)ξ(s)h(z)γ(dz, ds), t [,T], t [,T], t [,T]. In the next step we will calculate the difference ū 1 ū 2.Fix t T. imilarly as before we have ū 1 (t) ū 2 (t) =(Gū 1 )(t) (Gū 2 )(t) +(Hū 1 )(t) (Hū 2 )(t)+(f τm ū 1 )(t) (F τm ū 2 )(t). Note, that G and H are linear. In addition, (T (t)) t R is a unitary group on L 2 (R d ). Therefore, E Gu 1 (t) Gu 2 (t) 2 L E (ū 2 1 (s) ū 2 (s))g(z) 2 L ν(dz) ds. 2 Due to the integrability conditions on page 3, i.e. the integrability condition (i), and the fact that g is Lipschitz continuous, we know that for any v L 2 (R d ) we have vg(z) L 2ν(dz) C gc (ν) v L 2 and we can proceed E Gū 1 (t) Gū 2 (t) 2 L C gc 2 (ν) t E ū 1 (s) ū 2 (s) 2 L ds. 2 imilarly, we get for H by the Minkowski inequality and the Lipschitz continuity of h, E Hū 1 (t) Hū 2 (t) 2 L C 2 h t E (ū 1 (s) ū 2 (s))h(z) 2 L ν(dz) ds 2 Z CC h C (ν)t t E ū 1 (s) ū 2 (s) 2 L ds. 2

14 22 Page 14 of 31 A. de Bouard, E. Hausenblas, and M. Ondreját NoDEA The only term, which has to be carefully analysed is the nonlinear term given by (F τm ū 1 )(t) (F τm ū 2 )(t) = iλ t T (t s) ( ū 1 (s) α 1 ū 1 (s) ū 2 (s) α 1 ū 2 (s) ) 1 [,τm)(s) ds. Let γ and σ be given such that 1 σ = 1 γ, and 2 γ δ + d 2 < d σ(α 1), (3.12) 1 if d =1, > 1 if d =2, 2d d+2 if d>2. (3.13) Due to our assumption on α and δ, such a couple (γ,σ) exists. Let (γ,ρ) be an admissible pair, where γ is the conjugate exponent to γ, i.e. 2/ρ = d(1/2 1/γ). Let ρ be the conjugate exponent to ρ. Then, we have if d=1, 4 2 γ < if d=2, and 1 ρ 3 if d=1, < 2 if d=2, (3.14) 2d d 2 if d>2, 2 if d>2. Before continuing, let us shortly introduce the trichartz estimate. Let us define the convolution operator Tu(t) := t T (t r)u(r) dr, t. (3.15) Let (p,q ), (p 1,q 1 ) [2, ) [1, ) be two admissible pairs. Then for all T> we have by the trichartz estimate (see [19, p. 64]) sup t T Tu L q (,T ;L p ) C u L q 1 (,T ;L p 1 ). Applying the trichartz estimate we have, for any r 1, t T (t s) ( ū 2 (s) α 1 ū 2 (s) ū 1 (s) α 1 ū 1 (s) ) 1 [,τm)(s) ds ( ū 2 (s) α 1 ū 2 (s) ū 1 (s) α 1 ū 1 (s) ) 1 [,τm)(s) ρ r L ρ ([,T ];L γ ). By Hölder s inequality, recalling that σ is such that 1 σ = 1 γ,sothat 2 if d =1, σ > 2 if d =2, (3.16) d if d>2, ρ r L 2

15 NoDEA Uniqueness of the nonlinear chrödinger equation Page 15 of the right hand side of the above expression may be bounded by ( T ( ū2 (s) α 1 + ū 1 (s) α 1 )1 [,τm) ρ (ū1 (s) ū L σ 2 (s))1 ) r [,τm)(s) ρ ds L 2 ( T ( ū 2 (s) + ū 1 (s) )1 [,τm) (α 1)ρ (ū L (α 1)σ 1 (s) ū 2 (s))1 ) r [,τm)(s) ρ ds. L 2 By inequality (3.12) and obolev embeddings Theorems we know that H2(R δ d ) L σ(α 1) (R d ) continuously. This implies that the right hand side above is in turn bounded by ( T ( ū 2 (s) + ū 1 (s) )1 [,τm) (α 1)ρ (ū 1 (s) ū 2 (s))1 ) r [,τm)(s) ρ ds, L 2 and, using the Hölder inequality, by C(T ) sup ( ū2 (s) + ū 1 (s) )1 [,τm)(s) (α 1)ρ r s [,T ] T H δ 2 (ū1 (s) ū 2 (s))1 [,τm)(s) rρ ds. L 2 The definition of the stopping time allows to bound the above term by T 2C(T )m (α 1)ρ r (ū1 (s) ū 2 (s))1 [,τm)(s) rρ ds. L 2 Now, taking r =2/ρ, an application of the Grownwall Lemma gives E ū 1 (t) ū 2 (t) 2 L =. ince ū 2 1 and ū 2 are cádlág on L 2 (R d ), both processes ū 1 and ū 2 are indistinguishable in L 2 (R d ) on the time interval [,τ m ]. tep II: We show that P (τ m <T) asm. Observe that, thanks to the definition of τ m, {τ m <T} { u 1 L ([,T ];H2 δ) m or u 2 L ([,T ];H2 δ) m}. Therefore, P (τ m <T) P ( ) ( ) u 1 L ([,T ];H2 δ) m + P u 2 L ([,T ];H2 δ) m. H δ 2 ince u 1 and u 2 are cádlág in H2(R δ d ), and P-a.s. sup s T u 1 (s) H δ 2 sup s T u 2 (s) H δ 2 <, it follows ( ) P u i L ([,T ];H2 δ) m, < and as m,fori =1, 2. This implies P (τ m <T) asm. Hence, both processes u 1 and u 2 are undistinguishable on [,T]. Due to pathwise uniqueness, one can show that a unique strong solution exists. Theorem 3.8. If the conditions of Theorems 3.4 and 3.6 are satisfied, then there exists a unique strong solution to Eq. (3.2) in D([,T]; L 2 (R d )).

16 22 Page 16 of 31 A. de Bouard, E. Hausenblas, and M. Ondreját NoDEA Proof. Let δ < 1 be the constant given in Theorem 3.6. In order to apply Theorem 5.14 below, we put Y = H2 3d (R d ), Y = D(R d ), X = H2 1 (R d ), and X := D([,T]; L 2 (R d )). We have chosen Y = H2 3d (R d ), since Y can be arbitrarily large, we only have to be sure that the following points are satisfied and these points are satisfied by this choice. Let us define the functions a : [,T] X Y (t, x) T(t)x; b : [,T] [,T] X Y (t, s, x) T(t s)1 [,t) (s) F α (x)+ T (t s)1 [,t)(s) [xh(z)]ν(dz), in where F α (x) = x α 1 x,forx X L α+1 (R d ), so that F α (x) L (α+1)/α (R d ); c : [,T] [,T] X Y (t, s, z, x) T(t s)1 [,t) (s)(xg(z)); θ αi i : D([,T],Y) [, ], α i A i, i {, 1}, with A =[,T], A 1 =, { θ(u) if u D(,T; H δ = 2(R d )) if u D(,T; H2(R δ d )); if t (,T] then θ(u) t = if u D([,T]; L 2 (R d )) and, for u D([,T]; L 2 (R d )), t θ(u) t = { T (t s)f α (u(s)) L {y : T (t s)[u(s) G(y)] L 2 <1} {y : T (t s)[u(s) G(y)] L 2 1} + T (t s)[u(s) H(y)] L 2ν(dy)} ds where F α (x) isasabove. T (t s)[u(s) G(y)] 2 L 2 T (t s)[u(s) G(y)] L 2 Remark 3.9. Roughly speaking, the setting in Theorem 3.8 hastofittheset- ting in ect. 4 and has to be chosen as follows. The space X has to be a space such that for any t, the random variable u(t) isx valued. Note, that this space need not coincide with the space where the process is cádlág. The space Y has to be chosen such that the mappings a, b and c are well defined, D([,T]; Y ) corresponds to the Kurtz space (denoted in his paper by Z 1 ), moreover, X is the path space. Additional regularity properties of the solution can be incorporated in the family of functions {θ α : α A } and {θ α1 1 : α 1 A 1 }, where A and A 1 are index sets. In our case, we choose A =[,T]andA 1 =.

17 NoDEA Uniqueness of the nonlinear chrödinger equation Page 17 of Remark 3.1. The setting need not to be unique. Thus, instead of the setting above, we could choose X = H 1 2 (R d ), X = X = D([,T]; H δ 2 (R d )), A =, and A 1 =. Indeed, due to the fact that given pathwise uniqueness in L 2 (R d ) with the condition P ( u D([,T]; H δ 2 (R d ) ) = 1 one has pathwise uniqueness in H δ 2(R d ) and we could take directly X = D([,T]; H δ 2 (R d )). In the next step, we have to verify that the mappings a, b and c are measurable. In fact, first note that T is a strongly continuous unitary group on H2 3d (R d ), therefore, a is measurable. To show that b is measurable, we first investigate the measurability of T (t s)1 [,t) (s) F α (x). Here, we show that F α : H2 1 (R d ) Y is continuous, from which follows that F α : H2 1 (R d ) Y is measurable. Indeed, using Hölder inequality, it is easily seen that for any x, y L α+1 (R d ), ( ) F α (x) F α (y) L (α+1)/α (R d ) C x α 1 L α+1 (R d ) + y α 1 x y L α+1 (R d ) L α+1 (R ), d and the continuity result follows from the embeddings H2 1 (R d ) L α+1 (R d ) and L (α+1)/α (R d ) Y. ince 1 [,t) (s) is measurable, and (T (t)) t R is pointwise strongly continuous, we are done. It remains to show that the second term of b is measurable, but this follows since, by the Lipschitz continuity of h, and the fact that h() =, the Nemytsky operator H is continuous from L 2 (R d ) to Y. imilarly the last mapping c can be handled. It remains to verify that θ is a measurable mapping. But this is given, since D([,T]; H2 δ (R d )) is a Borel subset of D([,T]; Y ). Now, the existence of the strong solutions follows by an application of Theorem The abstract uniqueness result Let X and Y be separable Fréchet spaces, Y Y separates points in Y, X a Borel subset in D([,T],Y), A 1 and A 2 be two index sets and a :[,T] X Y, b :[,T] [,T] X Y, c :[,T] [,T] X Y, θ αi i : D([,T],Y) [, ], α i A i, i {, 1} be measurable mappings, ν a σ-finite measure on (, ), and, finally, n such that n and ν( n ) <. Let η be a time homogeneous Poisson random measure with intensity measure ν on the space (, ) defined over a probability space A =(Ω, F, F, P), where F denotes a filtration (F t ) t [,T ]. Given is an abstract evolution equation of the following form: t u(t),ϕ = a(t, u()),ϕ + b(t, s, u(s)),ϕ ds + c(t, s, x, u(s)),ϕ η(dx, ds), (4.1) for every t [,T]andϕ Y. We define next the concept of solution in the way we will use it in the following pages of the article.

18 22 Page 18 of 31 A. de Bouard, E. Hausenblas, and M. Ondreját NoDEA Definition 4.1. We say that a 6-tuple (Ω, F, F, P,u,η), F =(F t ) t [,T ],consisting of a filtered probability space A =(Ω, F, F, P), a time homogeneous Poisson random measure η on (, ) overa with intensity measure ν and a process u on [,T], being F-adapted and càdlàg in Y, is a solution of (4.1) provided that P(u(t) X) =1, t [,T], P (u X)=1, there exists p [1, 2] such that, for all t [,T], t b(t, s, u(s)),ϕ ds + + {x : c(t,s,x,u(s)),ϕ <1} {x : c(t,s,x,u(s)),ϕ 1} c(t, s, x, u(s)),ϕ p ν(dx) ds c(t, s, x, u(s)),ϕ ν(dx) ds < P-a.s. (4.2) hold for every t [,T], ϕ Y,andu solves Eq. (4.1). Additional regularity properties, which are not part of the definition of the solution, but which are essential for the pathwise uniqueness, are incorporated by the additional mappings θ α and θ α1 1, α A, α 1 A 1. Hypothesis 4.1. The solution u satisfies additionally P (θ α (u) < ) =1 and E θα1 1 (u) < for all α A, α 1 A 1. (4.3) Note that in the particular case of the Nonlinear chrödinger equation of ect. 3, {θ α } were defined in the proof of Theorem 3.8 and represent sufficient conditions for existence and uniqueness of the solution u. Definition 4.2. If η M N({ n R + }) then we define η t (V )=η(v ( (,t])), η t (V )=η(v ( (t, T ])), V B((,T]). (4.4) Lemma 4.3. If η is a time homogeneous Poisson random measure over a filtered probability space A =(Ω, F, F, P), then, for every t [,T], η t is an F t -measurable M N({ n R + })-valued random variable and η t is independent of F t. ince the proof is obvious, we omit it. Lemma 4.4. Let A =(Ω, F, F, P) be a filtered probability space with filtration F = (F t ) t [,T ], u a D([,T]; Y )-valued random variable over A, andη an M N({ n (,T]})-valued random variable over A. In addition, we assume that for any t, u(t) and η t are F t -measurable and η t is independent of F t. If there exists a solution A =( Ω, F, ( F t ) t, P, ū, η) of Eq. (4.1) satisfying Eq. (4.1) and the assumption of Definition 4.1, such that the law of (u, η) coincides with the law of (ū, η) on D([,T]; Y ) M N({ n [,T]}), then (Ω, F, (Ft P ) t, P,u,η) is a solution to Eq. (4.1) satisfying Hypothesis 4.1.

19 NoDEA Uniqueness of the nonlinear chrödinger equation Page 19 of Before giving the proof of Lemma 4.4, let us introduce some notations. For jointly Borel measurable mappings ă : R + X R, b : R + R+ X R and c : R + R+ Z X R, a time homogeneous Poisson random measure η with intensity measure ν and an adapted càdlàg process v in Y with v(t) X a.s. for every t R +, both defined on a probability space A =(Ω, F, F, P) with filtration F =(F t ) t [,T ], satisfying b(t, s, v(s)) Y ds + + {x : c(t,s,x,v(s)) Y <1} {x : c(t,s,x,v(s)) Y 1} c(t, s, x, v(s)) p Y ν(dx) ds c(t, s, x, v(s)) Y ν(dx) ds < for some p [1, 2] and every t (,T], define a nonlinear map K A a.s. K A (v, η)(t) =ă(t, v()) + b(t, s, v(s)) ds + c(t, s, x, v(s)) η(dx, ds). (4.5) Observe that K A (v, η) actually depends via the compensator of η, also on the probability measure P. Proof of Lemma 4.4. It is rather standard to prove that η is a time homogenous Poisson random measure with intensity ν for the augmented filtration (Ft P ) t, cf. Lemma 4.3, and that all the measure and integrability assumptions in Definition 4.1 are satisfied for P, u and η. o it just remains to prove that the actual equation (4.1) holds, i.e., that for any t R + and ϕ Y we have P (K A (u, η)(t) u(t),ϕ =)=1, where K A is defined with ă = a, ϕ, b = b, ϕ and c = c, ϕ ; Fix t R + and ϕ Y. Let us remind, that the mappings a :[,T] X Y, b :[,T] [,T] X Y, c :[,T] [,T] X Y, θ αi i : D([,T],Y) [, ], α i A i, i {, 1} are measurable. Hence, the mapping X v a t := a(t, v ),ϕ R is Borel measurable. ince u and ū belong a.s. to D([,T]; Y )andu(), ū() belong a.s. to X, u() and ū() have the same law on X, it follows by Lemma 1.22 [15] that the triplets (a t (u()),u,η) and (a t (ū()), ū, η) have the same law on X D([,T]; Y ) M N({ n (,T]}). ince b :[,T] [,T] X Y, is measurable, for any s [,T] u(s) and ū(s) are X-valued random variables, Law(ū(s)) = Law(u(s)), it follows

20 22 Page 2 of 31 A. de Bouard, E. Hausenblas, and M. Ondreját NoDEA that for any s, t [,T], the processes {b t (s) :s } and {b t (s) :s }, defined by b t (s, ω) := b(t, s, ū(s, ω),ϕ, s [,T], ω Ω, and b t (s, ω) := b(t, s, u(s, ω),ϕ, s [,T], ω Ω, have the same law for each t [,T]. In addition, we know by the definition of the solution that ( t ) ( t ) P b t (s) ds < =1 and P b t (s) ds < =1. By Theorem 8.3 of [21], it follows that for any t [,T] ( t ) ( t ) Law ū, b t (s)ds and Law u, b t (s)ds are equal on D([,T]; L 2 (R d )) Y. Finally, since c :[,T] [,T] X Y is measurable, for any s [,T], the random variable ū(s) is F s -measurable and the random variable u(s) isf s -measurable. It follows that the processes {c t (s) :s } and {c t (s) :s }, defined by c t (s, ω) := c(t, s, ū(s, ω),ϕ, s [,T], and c t (s, ω) := c(t, s, u(s, ω),ϕ, s [,T], are adapted to the filtrations ( F s ) s [,T ] and (F s ) s [,T ], respectively. Moreover ( ) P c t (x, s) p ν(dx) ds < =1 {x : c(t,s,x,ū(s)),ϕ <1} holds because ū is a solution, and since the law of (ū, η) coincides with the law of (u, η), we get that ( ) P c t (x, s) p ν(dx) ds < =1 {x : c(t,s,x,u(s)),ϕ <1} holds. Hence, P-a.s. the process [,T] s c t (s) R and P-a.s. the process [,T] s c t (s) R are belonging (but of the large jumps) to L p ([,T]; R). It follows by Proposition B.1(ii) that they are progressively measurable. Hence, Theorem A.1 is applicable and we know that (Ī(t), ū, η) and (I(t),u,η)have the same law on R D([,T]; Y ) M N({ n (,T]})), where T Ī(t) := c(t, s, ū(s),z) η(dz, ds), t [,T], and I(t) := T c(t, s, u(s),z) η(dz, ds), t [,T].

21 NoDEA Uniqueness of the nonlinear chrödinger equation Page 21 of and To deal with the large jumps, we use the fact that ( ) P c t (x, s) ν(dx) ds < =1, ( P {x : c t(x,s) 1} {x : c t(x,s) 1} ) c t (x, s) ν(dx) ds < =1. and proceed as above with p =1. umming up, it follows that, if (u, η) and(ū, η) have the same law on D([,T]; Y ) M N({ n (,T]}), then K Ā (ū, η)(t), ū, η) and (K A (u, η)(t),u,η) have the same law on Y D([,T]; Y ) M N({ n (,T]}). ince for all t [,T] P (K Ā (ū, η)(t) ū(t) =) =1, it follows that P (K A (u, η)(t) u(t) =)=1. In particular, the six tuple (Ω, P, F, F = (F t ) t,u,η) is a solution to (4.1). 5. Uniqueness Throughout this section, the notation of ect. 4 will be kept. We are going to prove here that the abstract result of Kurtz [17] can be applied to the problem (4.1). Or, in other words, that pathwise uniqueness for the Eq. (4.1) implies joint uniqueness in law and strong existence for the Eq. (4.1). In order to show this, let us define the Kurtz s compatibility structure and C-compatibility according to[17, Definition 3.3]. Throughout this section we fix an intensity measure ν and sets n such that n and ν( n ) <. Definition 5.1. Let us denote Z 1 = D([,T],Y), Z 2 = M N({ n (,T]}) X, Bt Z1 = σ(π s : s t), Bt Z2 = σ(r t ) B(X), where π t and R t are the canonical mappings, π t : D([,T],Y) π π(t) Y, R t : M N({ n (,T]}) M N({ n (,T]}) :μ μ( ( (,t])), and denote by C the Kurtz compatibility structure {(B Z1 t, Bt Z2 ):t [,T]}. Definition 5.2. If A is an Z 1 -valued random variable over some probability space A =(Ω, F, P), B a Z 2 -valued random variable over A and t [,T], then Ft A and Ft B are the coarsest σ-algebras such that the mappings A :(Ω, Ft A ) (Z 1, Bt Z1 ) and B :(Ω, Ft B ) (Z 2, Bt Z2 ) are measurable.

22 22 Page 22 of 31 A. de Bouard, E. Hausenblas, and M. Ondreját NoDEA Remark 5.3. If A is a Z 1 -valued random variable over A and B =(η, ξ) a Z 2 -valued random variable over A, it is rather standard to see that Ft A = σ(a s : s t) and Ft B = σ(η t ) σ(ξ) hold for every t [,T], as defined in (4.4). Definition 5.4. We say that Z 1 -valued random variables A 1,...,A n are C- compatible with a Z 2 -valued random variable B provided that E [h(b) F A1 t F An t F B t ]=E [h(b) F B t ] a.s. (5.1) holds for every t [,T] and every real bounded Borel measurable function h on Z 2. Remark 5.5. C-compatibility of random variables A 1,...,A n with a random variable B is actually a property of the joint law of (A 1,...,A n,b), as follows from [17, Remark 3.5]. Hence we can introduce the notion of C-compatibility for Borel probability measures on Z1 n Z 2, see [17, Definition 3.6] and Definition 5.6 below. Definition 5.6. A probability measure μ on Z1 n Z 2 is called C-compatible provided that, for any (A 1,...,A n,b) distributed according to μ, then A 1,...,A n are C-compatible with B in the sense of Definition 5.4. Lemma 5.7. Let η be a time homogeneous Poisson random measure with intensity measure ν on for some filtration (F t ) t [,T ] and let u be an F -measurable X-valued random variable. Then Z 1 -valued random variables A 1,...,A n are C-compatible with B =(η, u ) if and only if Ft A1 Ft An Ft B is P-independent of σ(η t ). Proof. ince η = η t + η t,(5.1) holds if and only if E [h(η t ) F A1 t F An t F B t ]=E [h(η t ) F B t ]. But σ(η t ) and Ft B = σ(η t ) σ(u ) are P-independent by Lemma 4.3, hence (5.1) holds if and only if Ft A1 Ft An Ft B is P-independent of σ(η t ). Now we are ready to define a Kurtz convexity constraint Γ ν, see [17, p. 958]. Definition 5.8. If μ is a Borel probability measure on Z 1 Z 2 and a random vector (u, η, u ) over a probability space (Ω, F, P) has the distribution μ, we say that μ satisfies a convexity constraint Γ ν provided that (a) u() = u almost surely, (b) there exists a solution ( Ω, F, F, ( F t ) t [,T ], P, ū, η) toeq.(4.1) satisfying Hypothesis 4.1 such that ν is the intensity measure of η and the law of (ū, η) coincides with the law of (u, η). Remark 5.9. Paradoxically, despite of the notion, we need not prove here that the Kurtz s convexity constraint Γ ν really defines a convex set of probability measures on Z 1 Z 2.Forus,Γ ν is viewed as a mere constraint with no convexity properties. We will explain more in the proof of Theorem 5.14.

23 NoDEA Uniqueness of the nonlinear chrödinger equation Page 23 of Finally, we define the Kurtz set Γν,C,Θ ν β, see [17, p. 958]. Definition 5.1. Let β be a Borel probability measure on X. We denote by Γν,C,Θ ν β the set of probability measures μ on Z 1 Z 2 such that (a) μ satisfies the convexity constraint Γ ν, (b) μ is C-compatible, (c) μ(z 1 )=Θ ν β on B(Z 2 ), where Θ ν is the law of the time homogeneous Poisson random measure with intensity measure ν, cf. Lemma 2.8. Remark Observe that the condition (b) in Definition 5.1 is redundant as it follows from (a) due to Lemma 5.7. We however present Definition 5.1 as it is, to be conformal with Kurtz s notation in [17]. Remark The Kurtz set Γν,C,Θ ν β is in fact convex. But since we do not need the convexity in this paper, we do not prove it. We can now give a full description of the set Γν,C,Θ ν β. Remark Let β be a Borel probability measure on X. Then μ Γν,C,Θ ν β if and only if there exists a solution ( Ω, F, F, ( F t ) t [,T ], P, ū, η) totheeq.(4.1) satisfying Hypothesis 4.1, ν is the intensity measure of η, μ is the law of (ū, η, ū()), β is the law of ū(). Proof. Comparing Definition 5.8 with Definition 5.1, taking into account Remark 5.11 and the fact that η is independent from ū() by Lemma 4.3, we get the equivalence immediately. Theorem Let β be a Borel probability measure on X and assume that there exists a solution ( Ω, F, F, ( F t ) t [,T ], P, ū, η) to Eq. (4.1) satisfying Hypothesis 4.1 such that ν is the intensity measure of η and β is the law of ū(), whenever (Ω, F, F, (F t ) t [,T ], P,u 1,η) and (Ω, F, F, (F t ) t [,T ], P,u 2,η) are solutions to Eq. (4.1) satisfying Hypothesis 4.1 such that ν is the intensity measure of η, β is the law of u 1 () and u 1 () = u 2 () a.s. then u 1 = u 2 a.s. Then there exists a Borel measurable mapping F : M N({ n (,T]}) X D([,T]; Y ) depending on ν and β such that (1) if (Ω, F, F, (F t ) t [,T ], P,u,η) is a solution to the Eq. (4.1) satisfying Hypothesis 4.1 such that ν is the intensity measure of η and β is the law of u() then u = F (η, u()) a.s. and u is adapted to the P-augmentation of the filtration (σ(η t ),σ(u())) t [,T ],

24 22 Page 24 of 31 A. de Bouard, E. Hausenblas, and M. Ondreját NoDEA (2) if (Ω, F, F, (F t ) t [,T ], P) is a stochastic basis, ξ is an X-valued F - measurable random variable with law β and η is a time homogeneous (F t ) t [,T ] -Poisson random measure with intensity ν then u = F (η, ξ) is (Ft P ) t [,T ] -adapted, u() = ξ a.s. and (Ω, F, F, (Ft P ) t [,T ], P,u,η) is a solution to (4.1) satisfying Hypothesis 4.1. Consequently, if (Ω i, F i, F i, (Ft i ) t [,T ], P i,u i,η i ), i =1, 2 are solutions to Eq. (4.1) satisfying Hypothesis 4.1 such that ν is the intensity measure of η 1 and η 2, β is the law of u 1 () and u 2 () then the law of (u 1,η 1 ) coincides with the law of (u 2,η 2 ). Proof. By the assumptions in Theorem 5.14, the Kurtz set Γν,C,Θ ν β is nonempty and pointwise uniqueness holds for C-compatible solutions of (Γ ν, Θ ν β) in the sense of [17, p. 959]. Hence by the implication (a) (b) in[17, Theorem 3.14], joint uniqueness in law holds for compatible solutions, i.e. Γν,C,Θ ν β contains exactly one measure, and there exists a strong compatible solution in the sense of [17, p. 959] and [17, Lemma 3.11], i.e. (1) holds. To prove (2), once u = F (η, ξ), we have that the law of (u, η, ξ) coincides with the law of (ū, η, ū()). Hence u() = ξ a.s. and u is compatible with (η, u()) by [17, Remark 3.5]. Thus u is adapted to the P-augmentation of the filtration (σ(η t ) σ(u())) t [,T ] by [17, Lemma 3.11]. The rest then follows from Lemma 4.4. We must point out here that the constraint Γ ν need not define a convex subset of Borel probability measures on Z 1 Z 2 if we apply just the implication (a) (b) in[17, Theorem 3.14]. The convexity of the Kurtz set Γν,C,Θ ν β is needed just for the implication (a) (b) in[17, Theorem 3.14] which we do not apply in our case. Publisher s Note pringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Appendix A: Uniqueness of the stochastic integral Let X and E be two separable Banach spaces. Later on we will take X to be one of the spaces E or L p (, ν, E). Let A =(Ω, F, (F t ) t [,T ], P) be an arbitrary filtered probability space and η be a Poisson random measure defined over A. LetN (Ω [,T]; X) be the space of (equivalence classes of) progressively measurable functions ξ :Ω [,T] X. For q (1, ) weset N q (Ω [,T], F; X) = M q (Ω [,T], F; X) = { ξ N(Ω [,T], F; X) : { ξ N(Ω [,T], F; X) :E ξ(t) q dt < a.s. } ξ(t) q dt <. }, (A.1) (A.2)