 → 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