Martingale solutions for the stochastic nonlinear Schrödinger equation in the energy space

We consider a stochastic nonlinear Schrödinger equation with multiplicative noise in an abstract framework that covers subcritical focusing and defocusing Stochastic NLSE in H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1$$\end{document} on compact manifolds and bounded domains. We construct a martingale solution using a modified Faedo–Galerkin-method based on the Littlewood–Paley-decomposition. For the 2d manifolds with bounded geometry, we use the Strichartz estimates to show the pathwise uniqueness of solutions.


Introduction
The article is concerned with the following nonlinear stochastic Schrödinger equation with a sequence of independent standard real Brownian motions (β m ) m∈N and functions (e m ) m∈N satisfying certain regularity and decay conditions that guarantee the convergence of the series on the RHS of (1.2) in the space E A .
The main aim of this study is twofold. Firstly, it proposes to construct a martingale solution of problem (1.1) by a stochastic version of a compactness method. Secondly, it proposes to prove the uniqueness of solutions by means of the stochastic Strichartz estimates. In this respect it differs from many previous papers on stochastic nonlinear Schrödinger equations, notably [8,18,28], and references therein, in which the proofs of both the existence and the uniqueness were obtained by means of appropriate stochastic Strichartz estimates. The compactness approach to the existence of solutions of 1-D stochastic Schrödinger equations in variational form has recently been used in a paper [31] by Keller and Lisei. Classical references for the construction of weak solutions of the deterministic NLSE by a combination of a compactness method and the Galerkin approximation are [23,24] for intervals and [42] as well as [56] for domains of arbitrary dimension. Let us point out that Burq et al. [4] also used a compactness method in the proof of their Theorem 3 but instead of the Galerkin approximation they used an approximation by more regular solutions. In particular, we give a new proof of these results. But we would like to emphasise that the deterministic case is significantly simpler since our spectral theoretic methods to construct the approximations of the noise term are not needed.
In technical sense, the present paper is motivated by the construction of a global solution of the cubic equation on compact 3d-manifolds M generalizing the existence part, see Theorem 3 of Burq et al. [4], to the stochastic setting. In three dimensions, the fixed point argument from [8] is restricted to higher regularity, because it requires the Sobolev embeddings H s,q → L ∞ , which are more restrictive in 3D than in 2D. Hence, this approach only yields local solutions, which is the motivation for constructing a global solution in H 1 (M) with an approximation procedure based on the conservation laws of the NLSE without using the dispersive properties of the Schrödinger group. We remark that in [4], the authors also prove uniqueness for the deterministic NLSE in 3D. For the equation with noise this question will be addressed in a forthcoming paper.
In the present paper, we construct a martingale solution of problem (1.1) by a modified Faedo-Galerkin approximation du n (t) = (−i Au n (t) − iP n F (u n (t))) dt − iS n B(S n u n (t)) • dW (t), t > 0, u n (0) = P n u 0 , (1.3) in finite dimensional subspaces H n of H spanned by some eigenvectors of A. Here, P n : H → H n are the standard orthogonal projections and S n : H → H n are selfadjoint operators derived from the Littlewood-Paley-decomposition associated to A. The reason for using the operators (S n ) n∈N lies in the uniform estimate sup n∈N S n L p →L p < ∞, 1 < p < ∞, which turns out to be necessary in the estimates of the noise due to the L p -structure of the energy, see (1.4) below, and which is false if one replaces S n by P n . Using the Littlewood-Paley decomposition via the operators (S n ) n∈N can be viewed as the one of the main analytical contributions of this paper. We remark that in the mean time, a similar construction has been used in [29] to construct a solution of a stochastic nonlinear Maxwell equation by estimates in L q for some q > 2. This indicates that our method has potential to increase the field of application of the classical Faedo-Galerkin method significantly. On the other hand, the orthogonal projections P n are used in the deterministic part, because they do not destroy the cancellation effects which lead to the mass and energy conservation for solutions u of problem (1.1) in the deterministic setting, whereF denotes the antiderivative of the nonlinearity F. Note that in the case F ± α (u) = ±|u| α−1 u, the antiderivative is given byF ± α = ± 1 α+1 u α+1 L α+1 . In the stochastic case, the mass conservation u n 2 L 2 = const for solutions of (1.3) holds almost surely due to the Stratonovich form of the noise. Moreover, the conservation of the energy is carried over in the sense that a Gronwall type argument yields the uniform a priori estimates, for every T > 0, where C w ([0, T ], E A ) denotes the space of continuous functions with respect to the weak topology in E A . The construction of a martingale solution is similar to [7] and employs a limit argument based on Jakubowski's extension of the Skorohod Theorem to nonmetric spaces and the Martingale Representation Theorem from [21, chapter 8].
For the details, we refer to Theorem 7.5 Let us point out that the stochastic nonlinear Schrödinger equations are used in the fiber optics, nonlinear photonics and optical wave turbulence, see for instance a recent review paper [51] by Turitsyn et al. and references therein. There is also an extended literature on the nonlinear Schrödinger equations on special manifolds, as e.g. Schwarzschild manifolds, see papers [1,38,40]. In these papers the Schrödinger equation is somehow related to the corresponding nonlinear wave equation which in turn appears in the theory of gravitational fields. Furthermore, we would like to mention the article [48] which deals with the derivation of the Schrödinger equation on manifolds. From a mathematical point of view, important questions are how the geometry of the manifold influences the qualitative behavior of solutions and how the geometry of the manifold and the external noise influence the well-posedness theory. Nonlinear Schrödinger equations on manifolds have been studied e.g. by Burg et al. [3,4], see also references therein. The motivation for these authors was "to evaluate the impact of geometry of the manifold on the well-posedness theory, having in mind the infinite propagation speed of the Schrödinger equation".
The paper is organized as follows. In the Sects. 2 and 3, we fix the notation, formulate our Assumptions and present a number of typical examples of operators A, a model nonlinearity F and noise coefficients B covered by our framework. In Sect. 4, we are concerned with the compactness results that we will be using later on. In Sect. 5, we formulate the Galerkin approximation equations and prove the a priori estimates which are sufficient for compactness in view of Sect. 4. Section 6 is devoted to the proof of Theorem 1 and in Sect. 7, we focus on uniqueness in the case of 2d manifolds with bounded geometry.
Let (X , , μ) be a σ -finite measure space with metric ρ satisfying the doubling property, i.e. μ(B(x, r )) < ∞ for all x ∈ X and r > 0 and μ(B(x, 2r )) μ(B(x, r )). (2.1) This estimate implies is called the energy space and the induced norm · E A is called the energy norm associated to A. We denote the dual space of E A by E * A and abbreviate the duality with ·, · := ·, · E * A ,E A , where the complex conjugation is taken over the second variable of the duality. Note that E A , H , E * A is a Gelfand triple, i.e.
(iv) Let α ∈ (1, p 0 − 1) be such that E A is compactly embedded in L α+1 (M). We set  where spectral multiplier theorems for S in L p (M) for p ∈ ( p 0 , p 0 ), respectively a Mihlin M β functional calculus of S for some β > 0 are employed. The Mihlin functional calculus is defined and studied in [32,34]. For additional information about spectral multiplier theorems for operators with generalized Gaussian estimates, we refer to [33,55]. Note that spectral multiplier results with different assumptions are also sufficient for our analysis below, see e.g. [20], where a result for the Laplace-Beltrami operator on a compact Riemannian manifold is explicitly stated without mentioning the doubling property in this particular case.
We start with some conclusions which can be deduced from Assumption 2.1.

Lemma 2.3 (a) There is a non-negative selfadjoint operatorÂ on E *
(c) There is an orthonormal basis (h n ) n∈N and a nondecreasing sequence (λ n ) n∈N with λ n > 0 and λ n → ∞ as n → ∞ and Proof (ad a) The operatorÂ is defined by The estimate shows thatÂ is well-defined and a bounded operator from E A to E * A with Â ≤ 1. Moreover, one can apply the Lax-Milgram-Theorem to see that I +Â is a surjective one can show the symmetry ofÂ as an unbounded operator in E * A . Hence,Â is selfadjoint, because −1 ∈ ρ(Â).
(ad b) The embedding E A → L α+1 (M) is compact by Assumption 2.1(iv) and L α+1 (M) → H is continuous due to μ(M) < ∞. Hence, E A → H is compact. (ad c) Immediate consequence of the spectral theorem, since S has a compact resolvent.
In most cases where this does not cause ambiguity or confusion, we also use the notations A forÂ. We continue with the assumptions on the nonlinear part of our problem.

Assumption 2.4
Let α ∈ (1, p 0 −1) be chosen as in Assumption 2.1. Then, we assume the following: Note that this leads to F: We further assume and F(0) = 0 and Re iu, F(u) = 0, u ∈ L α+1 (M). (2.6) The map F has a real antiderivativeF, i.e. there exists a Fréchet-differentiable mapF: By Assumption 2.4(ii) and the mean value theorem for Fréchet differentiable maps, we get x, y ∈ L α+1 (M), (2.9) which means that the nonlinearity is Lipschitz on bounded sets of L α+1 (M). We will cover the following two standard types of nonlinearities. Assumption 2. 6 We assume either (i) or (i ): (i) Let F be defocusing and satisfy u α+1 (2.10) (i ) Let F be focusing and satisfy Here (·, ·) θ,1 denotes the real interpolation space and we remark that by [54, Lemma 1.10.1], (2.12) is equivalent to for some β 1 > 0 and β 2 ∈ (0, 2) with α + 1 = β 1 + β 2 . Let us continue with the definitions and assumptions for the stochastic part.

Assumption 2.7
We assume the following: (i) Let ( , F, P) be a probability space, Y a separable real Hilbert space with ONB ( f m ) m∈N and W a Y -canonical cylindrical Wiener process adapted to a filtration F satisfying the usual conditions. and assume B m ∈ L(E A ) and B m ∈ L(L α+1 (M)) for m ∈ N and α ∈ (1, p 0 − 1) as in Assumption and Notation 2.1 with For the special case, when the B m are pointwise multiplication operators, see Sect. 3.5 below.
Finally, we have sufficient background to formulate the problem which we want to solve. We investigate the following stochastic evolution equation in the Stratonovich form where the stochastic differential is defined by For the purpose of giving a rigorous definition of a solution to problem (2.16), it is useful to rewrite the equation in the Itô form. Therefore, we first compute Hence, Eq. (2.16) will be understood in the following Itô form where the linear operator μ defined by is the Stratonovich correction term. Most of our paper will be concerned with the construction of a martingale solution.

Examples
In this section, we consider concrete situations and verify that they are covered by the general framework presented in the last section.

The model nonlinearities
The class of the general nonlinearities from the Assumptions 2.4 and 2.6 covers the standard focusing and defocusing power nonlinearity.
Then, F ± α satisfies Assumption 2.4 with antiderivativeF ± α . Furthermore, We can apply the Lemma 3.2 below with p = α + 1 and Then, the map

The Laplace-Beltrami operator on compact manifolds
In this subsection, we deduce Corollary 1.2 from Theorem 1.1. Let (M, g) be a compact d-dimensional Riemannian manifold without boundary and A := − g be the Laplace-Beltrami operator on M.
Proof of Corollary 1. 2 Step 1. Let X = M, ρ be the geodesic distance and μ be the canonical volume measure on X . From [16, Section 4, p. 329], we obtain the local doubling property of X , i.e. there is C 1 > 0 such that for all x ∈ X and r ∈ (0, 1) we have Dominated convergence implies that the function f : is continuous. Since X × [1, max{1, diam(M)}] is compact, we therefore obtain that In particular, this yields for every x ∈ X and r ∈ [1, max{1, diam(M)}]. For x ∈ X and r > diam(M), we get Step 2 Let S := I − g . Then, S is selfadjoint, strictly positive and commutes with A. Moreover, S has a compact resolvent and D(S k ) → E A holds for every k ∈ N.
Furthermore, S has upper Gaussian bounds by [25, Corollary 5.5 and Theorem 6.1], since these results imply for the kernel p of the semigroup e −t S t≥0 . This is sufficient for (2.4) since (2.2) implies In particular, S has generalized Gaussian bounds with p 0 = 1, see Remark 2.2. Next note that by Proposition B.2(a), the scale of Sobolev spaces on M is given by where the last identity can be deduced from the spectral theorem and (1+λ) s s 1+λ s . In particular, we have Then, by Proposition B.2(c) and Lemma 2.3, the embeddings are compact. Hence, Assumption 2.1 holds with our choice of A and S.
We have H 1 (M) → L p (M) and as above, interpolation between H and L p (M) yields Choosing ε small enough, we see that Assumption 2.6(i ) is true for α ∈ (1, 5).
Step 4 The Steps 1-3 and Theorem 1.1 complete the proof of Corollary 1.2.

Remark 3.3
Note, that the 3-dimensional case with a cubic defocusing nonlinearity, i.e.
is admissible in our framework. In the deterministic setting, i.e. B = 0, a global unique weak solution to this problem in H 1 (M) was constructed in [4, Theorem 3]. Uniqueness in the stochastic case will be proved in a forthcoming paper. In [8], the authors considered the stochastic problem, but only obtained global solutions in the 2-dimensional case.

Laplacians on bounded domains
We can apply Theorem 1.1 to the stochastic NLSE on bounded domains.

Corollary 3.4 Let M ⊂ R d be a bounded domain and be the Laplacian with Dirichlet or Neumann boundary conditions. In the Neumann case, we assume that ∂ M is
Lipschitz. Under Assumption 2.7 and either (i) or (ii)
We remark, that one could consider uniformly elliptic operators and more general boundary conditions, but for the sake of simplicity, we concentrate on the present two examples.
Proof In the setting of the second section, we choose X = R d . Hence, the doubling property is fulfilled. We consider the Dirichlet form a V : with associated operator (A V , D(A V )) in the following two situations: Hence, we obtain the same range of admissible powers α for the focusing and the defocusing nonlinearity as in the case of the Riemannian manifold without boundary.
In the Dirichlet case, we choose S := A = − D , which is a strictly positive operator and [46, Theorem 6.10], yields the Gaussian estimate for the associated semigroup. Hence, we can directly apply Theorem 1.1 to construct a martingale solution of problem (3.5).
In the Neumann case, we have 0 ∈ σ ( N ) and the kernel of the semigroup e −t N t≥0 only satisfies the estimate for all t > 0 and almost all (x, y) ∈ M × M with an arbitrary ε > 0, see [46,Theorem 6.10]. In order to get a strictly positive operator with the Gaussian bound from Remark 2.2, we fix ε > 0 and choose S := ε I − N . Finally, the computation of the admissible range of exponents α in the focusing case is similar to the third step of the proof of Corollary 1.2.

The fractional NLSE
In this subsection, we show how the range of admissible nonlinearities change when the Laplacians in the previous examples are replaced by their fractional powers (− ) β for β > 0. Exemplary, we treat the case of a compact Riemannian manifold without boundary. Similar results are also true for the Dirichlet and the Neumann Laplacian on a bounded domain. Let us point out that there exists a huge literature on the subject of fractional NLSE apparently starting with a paper [36] by Laskin. In the setting of Sect. 3.2, we look at the fractional Laplace-Beltrami operator given by A := − g β for β > 0, which is also a selfadjoint positive operator by the functional calculus and once again, we choose S := I − g . We apply Theorem 1.1 with The range of admissible pairs (α, β) in the defocusing case is given by since this is exactly the range of α and β with a compact embedding In the focusing case, analogous calculations as in the third step of the proof of Corollary 1.2 (with the distinction of β > d 2 , β = d 2 and β < d 2 ) imply that the range of exponents reduces to Hence, we get the following Corollary.

The model noise
In Corollaries 1.2 and 3.4, we considered the general linear noise from Assumption 2.7.
If M is either a compact Riemannian manifold or a bounded domain, let us consider the following example. Let (B m ) m∈N the multiplication operators given by We get Now, let d = 2 and q > 2 as in (3.8). Then, we have F → L ∞ (M). Furthermore, we choose p > 2 according to 1 2 = 1 q + 1 p and observe H 1 (M) → L p (M). As above, we obtain Hence, we conclude in both cases For d = 1, this inequality directly follows from the embedding The properties of B m as operator in L(L α+1 (M)) and in L(L 2 (M)) can be deduced from the embedding F → L ∞ (M). We close this section by remarks on natural generalizations of the linear, conservative noise considered in this paper. The details have been worked out in the second author's dissertation [27].

Remark 3.6
As in [8,Section 8], it is possible to replace the linear Stratonovich noise in Theorem 1.1, see also Assumption 2.7, by a nonlinear one of the form where we assume the Lipschitz and linear growth conditions In the case of H 1 -based energy spaces, i.e. the A = − on a bounded domain or A = − g on a Riemannian manifold, one can take g ∈ C 2 ([0, ∞), R) which satisfies the following conditions: This kind of nonlinearity is often called saturated and typical examples are given by for a constant σ > 0. For the Galerkin equation, we then take Unfortunately, this approximation does not respect mass conservation, but one still has which is enough for our purpose.

Remark 3.7
Another possible generalization of the noise is to drop the assumption that B m , m ∈ N, is selfadjoint. Then, the correction term μ has the form This kind of noise is called non-conservative and was considered in [12,28]. The existence result is then based on the approximation and the a priori estimates as well as the convergence results can be proved analogously.
We only have to replace mass conservation by the estimate (3.10). The uniqueness result in Sect. 7, however, only holds for selfadjoint B m , since this is the crucial assumption in Lemma 7.4.

Compactness and tightness criteria
This section is devoted to the compactness results which will be used to get a martingale solution of (1.1) by the Faedo-Galerkin method. Let A and α > 1 be chosen according to Assumption 2.1. We recall that the energy space E A is defined by E A := D(A 1 2 ). We start with a criterion for convergence of a Proof The Strauss-Lemma A.3 and the assumptions guarantee that By (a) and Banach-Alaoglu, we get a subsequence u n k k∈N and v ∈ L ∞ (0, and by the uniqueness of the weak star limit in . By Lemma A.2, we obtain the assertion.
We define a Banach spaceZ T bỹ and a locally convex space Z T by The latter is equipped with the Borel σ -algebra, i.e. the σ -algebra generated by the open sets in the locally convex topology of Z T . In the next Proposition, we give a criterion for compactness in Z T .

Proposition 4.2 Let K be a subset of Z T and r > 0 such that
Then, K is relatively compact in Z T .
Proof Let K be a subset of Z T such that the assumptions (a) and (b) are fullfilled and (z n ) n∈N ⊂ K . We want to construct a subsequence converging in Step 1 By (a), we can choose a constant C > 0 and for each n ∈ N a null set I n with z n (t) E A ≤ C for all t ∈ [0, T ]\I n . The set I := n∈N I n is also a nullset and A is also compact. Therefore, we can choose for each j ∈ N a Cauchy subsequence in E * A again denoted by z n (t j ) n∈N . By a diagonalisation argument, one obtains a common Cauchy subsequence z n (t j ) n∈N .
Let us choose finitely many open balls U 1 δ , . . . , U L δ of radius δ covering [0, T ]. By density, each of these balls contains an element of the sequence t j j∈N , say t j l ∈ U l δ for l ∈ {1, . . . , L} . In particular, the sequence z n (t j l ) n∈N is Cauchy for all l ∈ {1, . . . , L} . Hence, if we choose m, n ∈ N sufficiently large. Now, we fix t ∈ [0, T ] and take l ∈ {1, . . . , L} with |t j l − t| ≤ δ. We use (4.1) and (4.2) to get Step 2 The first step yields Step 3 We fix again ε > 0. By the Lions Lemma A.4 with X 0 = E A , X = L α+1 (M), for all v ∈ E A . The first step allows us to choose n, m ∈ N large enough that (4.4) and integration with respect to time yields Hence, the sequence (z n ) n∈N is also Cauchy in L α+1 (0, T ; L α+1 (M)).
In the following, we want to obtain a criterion for tightness in Z T . Therefore, we introduce the Aldous condition.

Definition 4.3
Let (X n ) n∈N be a sequence of stochastic processes in a Banach space E. Assume that for every ε > 0 and η > 0 there is δ > 0 such that for every sequence In this case, we say that (X n ) n∈N satisfies the Aldous condition [A].
The following Lemma (see [ The deterministic compactness result in Proposition 4.2 and the last Lemma can be used to get the following criterion for tightness in Z T .

Proposition 4.5 Let (X n ) n∈N be a sequence of continuous adapted E * A -valued processes satisfying the Aldous condition [A] in E *
A and Then the sequence P X n n∈N is tight in Z T , i.e. for every ε > 0 there is a compact set By Lemma 4.4, one can use the Aldous condition [A] to get a Borel subset A of This set K is compact by Proposition 4.2 and we can estimate In metric spaces, one can apply Prokhorov Theorem (see [47, Theorem II.6.7]) and Skorohod Theorem (see [5,Theorem 6.7]) to obtain convergence from tightness. Since the space Z T is a locally convex space, we use the following generalization to nonmetric spaces. are a subsequence μ n k k∈N , random variables X k , X for k ∈ N on a common probability space (˜ ,F,P) withP X k = μ n k for k ∈ N, and X k → XP-almost surely for k → ∞.
We stated Proposition 4.6 in the form of [9] (see also [30]) where it was first used to construct martingale solutions for stochastic evolution equations. We apply this result to the concrete situation and obtain the final result of this section. Then, there are a subsequence (X n k ) k∈N and random variablesX k ,X for k ∈ N on a second probability space (˜ ,F,P) withPX k = P X n k for k ∈ N, andX k →X P-almost surely in Z T for k → ∞.

The Galerkin approximation
In this section, we introduce the Galerkin approximation, which will be used for the proof of the existence of a solution to (1.1). We prove the well-posedness of the approximated equation and uniform estimates for the solutions that are sufficient to apply Corollary 4.7.
By the functional calculus of the selfadjoint operator S from Assumption and Notation 2.1, we define the operators P n : H → H by P n := 1 (0,2 n+1 ) (S) for n ∈ N 0 . Recall from Lemma 2.3, that S has the representation and observe that P n is the orthogonal projection from H to H n . Moreover, we have Note that we have h m ∈ k∈N D(S k ) for m ∈ N and thus, we obtain by the assumption D(S k ) → E A for some k ∈ N that H n is a closed subspace of E A for n ∈ N 0 . In particular, H n is a closed subspace of E * A . The fact that the operators S and A commute by Assumption 2.1 implies that P n and A 1 2 commute. We obtain and By density, we can extend P n to an operator P n : Despite their nice behaviour as orthogonal projections, it turns out that the operators P n , n ∈ N, lack the crucial property needed in the proof of the a priori estimates of the stochastic terms. In general, they are not uniformly bounded from L α+1 (M) to L α+1 (M). To overcome this deficit, we construct another sequence (S n ) n∈N of operators S n : H → H n using functional calculus techniques and the general Littlewood-Paley decomposition from [34].
In the next Proposition, we use the estimate from Lemma 5.1 to construct the sequence (S n ) n∈N which we will employ in our Galerkin approximation of the problem (1.1). For a more direct proof which employs spectral multiplier theorems from [33,55] rather than the abstract Littlewood-Paley theory from [34], we refer to [27]. Moreover, we would like to remark that in the meantime, a similar construction has also been applied to use the Galerkin method in the context of stochastic Maxwell equation, see [29]. is selfadjoint for each m, since ρ m is real-valued. Hence, S n is selfadjoint. By the convergence property of the functional calculus, we get S n ϕ → ϕ in E A for all ϕ ∈ E A . A straightforward calculation using the properties of the dyadic partition of unity leads to Using the operators P n and S n , n ∈ N, we approximate our original problem (1.1) by the stochastic differential equation in H n given by With the Stratonovich correction term the approximated problem can also be written in the Itô form du n (t) = (−i Au n (t) − i P n F (u n (t)) + μ n (u n (t))) dt − iS n B(S n u n (t))dW (t), u n (0) = P n u 0 .
By the well known theory for finite dimensional stochastic differential equations with locally Lipschitz coefficients, we get a local wellposedness result for (5.5). almost surely for all t ≥ 0.

Proof
Step 1 We fix n ∈ N and take the unique maximal solution (u n , τ n ) from Proposition 5.3. We show that the estimate (5.6) holds almost surely on {t ≤ τ n }. The function Φ: For the sequence τ n,k k∈N of stopping times we have τ n,k τ n almost surely and the Itô process u n has the representation

S n B(S n u n (s))dW (s)
almost surely on {t ≤ τ n,k } for all k ∈ N. We fix k ∈ N.
where we used (5.2) and Assumption 2.4(i) for the second term and the fact, that the operator S n B m S n is selfadjoint for the third term. Analogously, we get Thus, we obtain u n (t) 2 H = P n u 0 2 H ≤ u 0 2 H almost surely in {t ≤ τ n,k }.
Step 2 To show τ n = ∞ almost surely, we assume the contrary. Therefore, there is 0 ∈ F with P( 0 ) > 0 such that τ n (ω) < ∞ and τ n,k (ω) τ n (ω) for all ω ∈ 0 . Hence, τ n,k < ∞ on 0 and by the continuity of the paths of u n and the definition of τ n,k , we get u n (τ n,k (ω), ω) H n = k for all ω ∈ 0 and k ∈ N. This is a contradiction to Step 1, where we obtained u n (t) H ≤ u 0 H almost surely in {t ≤ τ n,k }. Therefore, u n is a global solution and we have The next goal is to find uniform energy estimates for the global solutions of the Eq. (5.5). Recall that by Assumption 2.4, the nonlinearity F has a real antiderivative denoted byF.

Definition 5.5 We define the energy E(u) of u ∈ E A by
Note that E(u) is welldefined by the embedding E A → L α+1 (M). In contrast to the uniform L 2 -estimate in [0, ∞), we cannot exclude the growth of the energy in an infinity time interval. So, we fix T > 0 from now on. As a preparation, we formulate a Lemma, which simplifies the arguments, when the Burkholder-Davis-Gundy inequality is used.

Lemma 5.6
Let r ∈ [1, ∞), ε > 0, T > 0 and X ∈ L r ( , L ∞ (0, T )). Then, Proof By interpolation of L 2 (0, t) between L ∞ (0, t) and L 1 (0, t) and the elementary inequality √ ab ≤ εa + 1 4ε b for a, b ≥ 0 and ε > 0, we obtain Now, we take the L r ( )-norm and apply Minkowski's inequality to get The next Proposition is the key step to show that we can apply Corollary 4.7 to the sequence of solutions (u n ) n∈N of the Eq. (5.5) in the defocusing case.

Proposition 5.7
Under Assumption 2.6(i), the following assertions hold.
In particular, for all r ∈ [1, ∞) there is Proof (ad a) By Assumption 2.4(ii) and (iii), the restriction of the energy E: H n → R is twice continuously Fréchet-differentiable with Re Au n (s) + F(u n (s)), μ n (u n (s)) ds Re Au n (s) + F(u n (s)), −iS n B (S n u n (s)) dW (s) Re F [u n (s)] (S n B m S n u n (s)) , S n B m S n u n (s) ds (5.7) almost surely for all t ∈ [0, T ]. We can use (5.2) for for all v ∈ H n to simplify (5.7) and get almost surely for all t ∈ [0, T ]. Next, we fix δ > 0, q > 1 and apply the Itô formula to the process on the LHS of (5.8) and the function Φ: The derivatives are given by With the short notation we obtain Re Au n (s) + F(u n (s)), μ n (u n (s)) ds

u n (s) H S n B m S n u n (s) E
(5.14) Substituting the inequalities (5.12) to (5.16), into the identity (5.9), we get for each Choosing ε > 0 small enough in inequality (5.17), the Gronwall lemma yields with a constant C > 0, which is uniform in n ∈ N. Because of we obtain the assertion of Proposition 5.7, part (a).
in H n almost surely for all t ∈ [0, T ] and therefore for each sequence (τ n ) n∈N of stopping times and θ > 0. Hence, we get for a fixed η > 0. We aim to apply Tschebyscheff's inequality and estimate the expected value of each term in the sum. We use part a) for the embedding L α+1 α (M) → E * A and the estimate (2.5) of the nonlinearity F for Propositions 5.2 and 5.4 for Finally, we use the Itô isometry and again the Propositions 5.2 and 5.4 for By the Tschebyscheff inequality, we obtain for a given η > 0 for k ∈ {1, 2, 3} and (5.20) Let us fix ε > 0. Due to estimates (5.19) and (5.20) we can choose δ 1 , . . . , δ 4 > 0 such that for 0 < θ ≤ δ k and k = 1, . . . , 4. With δ := min {δ 1 , . . . , δ 4 } , using (5.18) we get for all n ∈ N and 0 < θ ≤ δ and therefore, the Aldous condition [A] holds in E * A . We continue with the a priori estimate for solutions of (5.5) with a focusing nonlinearity. Note that this case is harder since the expression does not dominate v 2 E A , becauseF is negative. Proposition 5.8 Under Assumption 2.6(i ), the following assertions hold: Proof Let ε > 0. Assumption 2.6(i ) and Young's inequality imply that there are γ > 0 and C ε > 0 such that (5.21) and therefore by Proposition 5.4, we infer that By the same calculations as in the proof of Proposition 5.7 we get Re Au n (r ) + F(u n (r )), μ n (u n (r )) dr Re Au n (r ) + F(u n (r )), −iS n B (S n u n (r )) dW (r ) Re F [u n (r )] (S n B m S n u n (r )) , S n B m S n u n (r ) dr (5.23) almost surely for all s ∈ [0, T ]. In the following, we fix q ∈ [1, ∞) and t ∈ (0, T ] and want to apply the L q ( , L ∞ (0, t))-norm to the identity (5.23). We will use the notation For the following estimates, we will use (5.13)-(5.16) and the Minkowski inequality and obtain · 0 Re Au n (s) + F(u n (s)), μ n (u n (s)) ds

s)] (S n B m S n u n (s)) , S n B m S n u n (s) ds
(5.29) By (5.23) and the estimates (5.25)-(5.29), we get In order to estimate the terms with X by the LHS of (5.30), we exploit (5.21) to get ∞ (0,t) Hence, by (5.24), we obtain Choosing ε > 0 small enough, we get for all t ∈ [0, T ] and thus, the Gronwall Lemma yields This implies that there is C > 0 with since the H -norm is conserved by Proposition 5.4. Therefore, we obtain the assertion for r ≥ 2. Finally, the case r ∈ [1, 2) is an application of Hölder's inequality.
(ad b) Analogous to the proof of Proposition 5.7(b).

Construction of a martingale solution
The aim of this section is the construction of a solution of Eq. (1.1) by a suitable limiting process in the Galerkin equation (5.5) using the results from the previous sections. Let us recall that (a) There are a subsequence u n k k∈N , a probability space ˜ ,F ,P and random H k )P-a.s. and for all r ∈ [1, ∞), there is C > 0 with .
For the precise dependence of the constants, we refer to the Propositions 5.7 and 5.8.

Proof
Use the Propositions 5.7 in the defocusing respectively 5.8 in the focusing case to get the assertion. for n → ∞. On the other hand, part b) yields the existence ofṽ ∈ L r (˜ , L ∞ (0, T ; E A )) for all r ∈ [1, ∞) with norm less than the constant C = C( u 0 E A , T , r ) > 0 and a subsequence v n k k∈N , such that v n k * ṽ for k → ∞. Especially, v n k * ṽ for k → ∞ in L 2 (˜ , L α+1 (0, T ; L α+1 (M))) and hence, v =ṽ ∈ L r (˜ , L ∞ (0, T ; E A )).
The next Lemma shows, how convergence in Z T can be used for the convergence of the terms appearing in the Galerkin equation. Lemma 6.2 Let z n ∈ C([0, T ], H n ) for n ∈ N and z ∈ Z T . Assume z n → z for n → ∞ in Z T . Then, for t ∈ [0, T ] and ψ ∈ E A as n → ∞

Proof
Step 1 We fix ψ ∈ E A and t ∈ [0, T ]. Recall, that the assumption implies z n → z for n → ∞ in C([0, T ], E * A ). This can be used to deduce By z n → z in C w ([0, T ], E A ) we get sup s∈[0,T ] | z n (s) − z(s), ϕ | → 0 for n → ∞ and all ϕ ∈ E * A . We plug in ϕ = Aψ and use Az n (s), ψ = z n (s), Aψ for n ∈ N and s ∈ [0, t] to get Step 2 First, we fix m ∈ N. Using that the operators B m and S n are selfadjoint, we get and Lebesgue's convergence Theorem, we obtain and therefore t 0 μ n (z n (s)) , ψ H ds → t 0 μ (z(s)) , ψ ds, n → ∞.
Step 3 Before we prove the last assertion, we recall z n → z in L α+1 (0, T ; L α+1 (M)) for n → ∞. We estimate t 0 P n F(z n (s)), ψ H − F(z(s)), ψ ds where we used (5.2). For the first term in (6.1), we look at This leads to the last claim.
By the application of the Skorohod-Jakubowski Theorem, we have replaced the Galerkin solutions u n by the processes v n on˜ . Now, we want to transfer the properties given by the Galerkin equation (5.5). Therefore, we define the process for n ∈ N and t ∈ [0, T ] and in the following lemma, we prove its martingale property. Note that in this section, we consider H as a real Hilbert space equipped with the real scalar product Re u, v H for u, v ∈ H in order to be consistent with the martingale theory from [21] we use. We define a process N on˜ × [0, T ] by By Proposition 6.1, we infer that v ∈ C([0, T ], E * A ) almost surely and In the next Lemma, we use the martingale property of N n for n ∈ N and a limiting process based on Proposition 6.1 and Lemma 6.2. to conclude that L N is also an H -valued martingale.
By the Davis inequality for continuous martingales (see [49]), Lemma 6.3 and Proposition 6.1, we concludẽ E sup Step 2 Let ψ, ϕ ∈ E A and h be a bounded continuous function on C([0, T ], E * A ). For 0 ≤ s ≤ t ≤ T , we define the random variables for a, b ≥ 0 and p ≥ 1 and the estimate (6.2) for In view of the Vitali Theorem, we get Step 3 For 0 ≤ s ≤ t ≤ T , we define By Lemma 6.2, we obtain g 1,n (t, s) → g 1 (t, s)P-a.s. for all 0 ≤ s ≤ t ≤ T . In order to get uniform integrability, we set r := α+1 2 > 1 and estimatẽ where we used (6.2) again. As above, Vitali's Theorem yields Step 4 For 0 ≤ s ≤ t ≤ T , we define The first and the third term tend to 0 as n → ∞ by Proposition 6.1 and for the second one, this follows by the estimate and Lebesgue's convergence Theorem. Hence, we conclude and continue with r := α+1 2 > 1 and Using Vitali's Theorem, we obtain Step 5 From step 2, we havẽ and step 3, step 4 and Lemma 6.3 yield Now, let η, ζ ∈ H . Then ι * η, ι * ζ ∈ E A and for all z ∈ E * A , we have Re Lz, η H = Re z, ι * η . By the first step, L N is a continuous, square integrable process in H and the identities (6.3) and (6.4) implỹ A and adapted to the filtrationF given bỹ Hence, Φ is continuous in H and adapted tõ F and therefore progressively measurable. By an application of Theorem 8.2 in [21] to the process L N from Lemma 6.4, we obtain a cylindrical Wiener processW on Y defined on a probability space and using the continuity of the operator L, we get It remains to prove the mass conservation from Theorem 1.1. In Proposition 5.4, we proved a similar result for the approximating equation. Since this property is not invariant under the limiting procedure from above, we have to repeat the calculation in infinite dimensions and justify it by a regularization procedure.  Re R λ u(s), R λ μ(u(s)) H → Re u(s), μ(u(s)) H , λ → ∞. (6.10) by (6.6). In order to apply the dominated convergence Theorem by Lebesgue, we estimate using (6.6) and the Sobolev embeddings L α+1 by Lebesgue. The Itô isometry and the Doob inequality yield After passing to a subsequence, we get Step 3 Using (6.9), (6.11) and (6.12) in (6.8), we obtain

Regularity and uniqueness of solutions on 2d manifolds
In this section, we want to study pathwise uniqueness of solutions to (1.1) and we consider the case of a 2-dimensional Riemannian manifold without boundary M. We drop the assumption that M is compact and replace it by M is complete, has a positive injectivity radius and a bounded geometry. (7.1) We refer to [53, chapter 7], for the definitions of the notions above and background references on differential geometry. We equip M with the canonical volume μ and suppose that M satisfies the doubling property: for all x ∈M and r > 0, we have μ (B(x, r )) < ∞ and μ(B(x, 2r )) μ (B(x, r )).
We emphasize that (7.1) is satisfied by compact manifolds. Examples for manifolds with the property (7.2) are given by compact manifolds and manifolds with nonnegative Ricci-curvature, see [16]. Let A = − g be the Laplace-Beltrami operator F = F ± α be the model nonlinearity from Sect. 3. The proof is based on an additional regularity of the solution, which we obtain by applying the deterministic and the stochastic Strichartz estimates from [8,13].
In two dimensions, the mapping properties of the nonlinearity improve, as we will see in the first Lemma.

Proof
Step 1 First, we consider the case s = 1. Take q ∈ [2, ∞) and r ∈ (2, ∞) with The condition (7.3) yieldss and therefore, the assertion follows by applying the Sobolev embedding H 1,r (M) → Hs(M).  Proof of Proposition 7. 2 Step 1 First, we will show that it is possible to rewrite the Eq. (2.19) from the definition of solutions for (1.1) in the mild form (7.8). We note that for each s 0 < 0 the semigroup e −it A t≥0 on L 2 (M) extends to a semigroup T s 0 (t) t≥0 with the generator A s 0 that extends A to D(A s 0 ) = H s 0 +2 (M). To keep the notation simple, we also call this semigroup e −it A t≥0 . We apply the Itô formula to Φ ∈ C 1,2 ([0, t] × H s−2 (M), H s−4 (M)) defined by and obtain Step 2 Using the Strichartz estimates from Lemma B.4 we deal with the free term and each convolution term on the right hand site to get (7.7) and the identity (7. Integration over˜ and (7.6) yields To estimate the other convolutions, we need that μ is bounded in Hs(M) and B is bounded from Hs(M) to HS(Y , Hs(M)). This can be deduced from the following estimate, which follows from complex interpolation (see [  The estimates (B.7), (7.9) and (7.6) imply Hence, the mild Eq. (7.8) holds almost surely in Hs(M) for each t ∈ [0, T ] and thus, we get (7.7) by the pathwise continuity of deterministic and stochastic integrals.
As a preparation for the proof of pathwise uniqueness, we show a formula for the L 2 -norm of the difference of two solutions of (1.1).
Proof The proof is similar to Proposition 6.5. In fact, it is even simpler, since the regularity of F ± α due to Lemma 7.1 simplifies the proof of the convergence for λ → ∞.
Finally, we are ready to prove the pathwise uniqueness of solutions to (1.1).

Remark 7.7
A similar Uniqueness-Theorem can also be proved on bounded domains in R 2 using the Strichartz inequalities by Blair, Smith and Sogge from [14]. We also want to mention the classical strategy by Vladimirov (see [15,43,45,56]) to prove uniqueness of H 1 -solutions using Trudinger type inequalities which can be seen as the limit case of Sobolev's embedding, see also [2,Theorem 8.27]. Since this proof only relies on the formula (7.10) and the property of solutions to be in H 1 , it can be directly transfered to the stochastic setting. This strategy does not use Strichartz estimates, but it suffers from a restriction to α ∈ (1, 3] and it cannot be transfered to H s for s < 1.
Now, we give the definition of the concepts of strong solutions and uniqueness in law used in Corollary 1.3. The weak topology on B r X is metrizable if the dual X * is separable and a metric is given by for a dense sequence x * k k∈N ∈ B 1 X * N , see [11], Theorem 3.29. If X is also We continue with some auxiliary results.