Uniqueness of martingale solutions for the stochastic nonlinear Schr\"odinger equation on 3d compact manifolds

We prove pathwise uniqueness for solutions of the nonlinear Schr\"{o}dinger equation with conservative multiplicative noise on compact 3D manifolds. In particular, we generalize the result by Burq, G\'erard and Tzvetkov (N. Burq, P. G\'erard, and N. Tzvetkov. Strichartz inequalities and the nonlinear Schr\"{o}dinger equation on compact manifolds. American Journal of Mathematics, 126 (3):569--605, 2004) to the stochastic setting. The proof is based on deterministic and stochastic Strichartz estimates and the Littlewood-Paley decomposition.


INTRODUCTION AND MAIN RESULT
This article is concerned with the nonlinear Schrödinger equation with multiplicative noise In the previous article [9], we constructed a martingale solution of (1.1) in arbitrary dimension for λ = 1 and α ∈ (1, 1 + 4 (d−2) + ) or λ = −1 and α ∈ (1, 1 + 4 d ). Moreover, we proved pathwise uniqueness of solutions in the 2D-case. The aim of the present article is to show pathwise uniqueness in the significantly harder three dimensional case and to generalize the result by Burq, Gérard and Tzvetkov from [11], Theorem 3, for the cubic NLS to the stochastic setting. Then, solutions of (1.1) are pathwise unique.
Note that in contrast to existence, the uniqueness result is the same for the focusing and defocusing NLS. As an immediate consequence of the Yamada-Watanabe-Theory developed in [24], Theorem 5.3 and Corollary 5.4, we obtain the existence of a unique strong solution of (1.1).
The question of existence and uniqueness of global solutions of the stochastic nonlinear Schrödinger equation was previously addressed by de Bouard and Debussche in [14] and [15], Barbu, Röckner and Zhang in [1], [2], [33] and Hornung in [20]. In these articles, the authors considered the fullspace R d and employed a fixed point argument based on Strichartz estimates to prove existence and uniqueness in one step. As in the deterministic setting, their ranges of exponents α depend on the space dimension and the considered regularity. Brzeźniak and Millet followed a similar approach for the stochastic NLS on a compact 2D manifold M. In higher dimensions, their argument only yields local solutions since the estimates for the nonlinearity rely on the Sobolev embeddings H s,p ֒→ L ∞ that are too restrictive to work in the energy space H 1 (M). Another result about the stochastic NLS is due to Keller and Lisei, see [22], who considered the equation on the space-interval (0, 1) with Neumann boundary conditions. They proved existence with a Galerkin method and uniqueness via the Sobolev embedding H 1 (0, 1) ֒→ L ∞ (0, 1). Hence, their argument cannot be transfered to higher dimensions. After this work was finished, we learned about a recent paper [12] by Cheung and Mosincat. Using the additional structure in the special case of the d-dimensional torus M = T d and algebraic nonlinearities, i.e. α = 2k + 1 for some k ∈ N, the authors employed a fixed point argument based on multilinear Strichartz estimates and an estimate of the stochastic convolution in Bourgain spaces X s,b combined with the truncation method from [14], [15] and [20]. As a result, they solved the NLS with multiplicative noise in ) for all s > s crit := d 2 − 2 α−1 and some b < 1 2 as well as some stopping time τ > 0. As a byproduct, their argument also implies pathwise uniqueness of martingale solutions in L 2 (Ω, C([0, T ], H s (T 3 )) ∩ X s,b ([0, T ])) for α = 3 and s > 1 2 , which reflects an improvement compared to the general case considered in Theorem 1.1.
Our approach separates existence and uniqueness. The construction of a martingale solution in [9] did not use Strichartz estimates. It was only based on the Hamiltonian structure of the NLS and the compactness of the embedding H 1 (M) ֒→ L p (M). Since these ingredients are independent of the underlying geometry, the proof worked in a more general framework. In particular, we considered arbitrary dimensions d ∈ N and powers α ∈ (1, 1+ 4 (d−2) + ) and could also deal with Dirichlet and Neumann Laplacians on bounded domains as well as their fractional powers. The flexibility of this approach is underlined by the fact that it could be also used to construct a martingale solution of the NLS with pure jump noise, see [8]. In the following, we would like to explain the difficulties of the uniqueness result in the three dimensional case and sketch the proof, which is inspired by the ideas of Burq, Gérard and Tzvetkov in [11]. We take two solutions with u 1 , u 2 ∈ L ∞ (0, T ; H 1 (M)) almost surely. Our starting point is the representation for small times T h and the global Strichartz estimate from [11] for p, q ∈ [2, ∞] with 2 q + d p = d 2 and (q, p, d) = (2, ∞, 2). Here, h ∈ (0, 1] and ϕ ∈ C ∞ c (R) can be chosen arbitrarily. In two dimensions, (1.5) improves the regularity to u 1 , u 2 ∈ L q (0, T ; H s− 1 q ,p ) almost surely for s ∈ (1 − 1 q , 1). Hence, one can use a Gronwall argument based on the Sobolev embedding H s− 1 q ,p (M) ֒→ L ∞ (M) to prove pathwise uniqueness. For the details, we refer to [9]. In 3D, the challenge is to gain 1 2 + ε derivatives with respect to the embedding H 3 2 +ε (M) ֒→ L ∞ (M) in order to control the nonlinearity in (1.3) by the H 1 -estimates of the solutions. Unfortunately, this is not possible, but it turns out that one can replace L ∞ -estimates by for all p ∈ [6, ∞) and intervals J ⊂ [0, T ]. Then, we use (1.6) and the control of the L p -norms for 2 ≤ p ≤ 6 by H 1 (M) ֒→ L 6 (M) to get an inequality In order to get (1.6), we use partitions of unity to estimate the solutions locally in time and frequency by the Strichartz estimate (1.4). To control the stochastic term, we adapt Brzeźniak's and Millet's approach from [10] to derive a spectrally localized stochastic Strichartz estimate. Afterwards, we reassemble the local estimates by Littlewood-Paley-Theory. We point out that the proof is restricted to dimension d = 3 and α ∈ (1,3]. In fact, we need the endpoint Strichartz estimate by Keel and Tao, [21], to prove pathwise uniqueness for α = 3. We would like to point out that recently, Bernicot and Savoyeau, see [3], could prove estimates of the type of (1.4) and (1.5) also in the case of possibly non-compact manifolds with bounded geometry. Unfortunately, their estimate (1.4) only holds for T ≤ h 1+ε and (1.5) holds with loss 1+ε p for an arbitrary ε > 0. Moreover, the constants depend on ε, which leads to an additional growth of the constant in (1.6) as p → ∞. Hence, the results from [3] cannot be applied scheme of proof.
The strategy to use estimates of the type (1.7) to prove uniqueness was developed by Yudovitch, [32], for the Euler equation. In the context of the NLS, it was used by Vladimirov in [31], Ogawa and Ozawa in [26] and [27]. They looked at 2D domains and used Trudinger type inequalities as an analogon to (1.6) to control the growth of L p -norms for p → ∞. Burq, Gérard and Tzvetkov could use the Yudovitch-strategy for three dimensional manifolds without boundary due to the regularizing effect of Strichartz estimates. In [4], Blair, Smith and Sogge proved uniqueness of weak solutions of the deterministic NLS on compact 3D manifolds with boundary as an application of their Strichartz estimates on this type of geometry.
The paper is organized as follows. In section 2, we fix the notations, formulate our assumptions and collect auxiliary results. Section 3 is devoted to proof of the estimate (1.6) and the pathwise uniqueness.

DEFINITIONS AND AUXILIARY RESULTS
This section is devoted to the notations, definitions and auxiliary results that will be used in the next section to show pathwise uniqueness.
If a, b ≥ 0 satisfy the inequality a ≤ Cb with a constant C > 0, we write a b. Given a b and b a, we write a b. For two Banach spaces E, F , we denote by L(E, F ) the space of linear bounded operators B : E → F and abbreviate L(E) := L(E, E). We use the notation HS(H 1 , H 2 ) for the space of Hilbert-Schmidt-operators between Hilbert spaces H 1 and H 2 .
Let M be a three dimensional compact riemannian C ∞ manifold without boundary and L p (M) for p ∈ [1, ∞] the space equivalence classes of C-valued p-integrable functions. The distance induced by g is denoted by ρ and canonical measure on M is called µ. By L p (M) for p ∈ [1, ∞], we denote the space of equivalence classes of C-valued p-integrable functions with respect to µ. The Laplace-Beltrami operator on M, i.e. the generator of the heat semigroup on M, is named ∆ g . Moreover, we use the fractional Sobolev spaces For all s ∈ R, we shortly denote H s (M) := H s,2 (M). For properties of the Laplace-Beltrami operator, characterizations of the fractional Sobolev spaces and embedding theorems, we refer to [29] and [28]. For s = 1, one can show that the definition from above coincides with the classical Sobolev space and u 2 L 2 + ∇u 2 L 2 1 2 defines an equivalent norm on H 1 (M). We refer to [25] for an explanation of the gradient as an element of the tangential bundle of M.
Next, we summarize the assumptions on the coefficient of the noise in (1.1).
and that B m is symmetric as operator in In particular, we have B ∈ L (H 1 (M), HS(Y, H 1 (M))) and µ ∈ L(H 1 (M)) if we abbreviate We look at the following slight generalization of (1.1) in the Itô form In the introduction, we used that the process with a sequence (β m ) m∈N of independent Brownian motions is a cylindrical Wiener process in Y, see [13], Proposition 4.7, and the identity which relates Itô and Stratonovich noise. For the sake of simplicity, we restricted ourselves to the special case of multiplication operators We want to justify that they fit in Assumption 2.1. The Sobolev embedding H 1 (M) ֒→ L 6 (M) and the Hölder inequality yield Thus, Note that the existence-Theorem from [9] additionally needs the assumptions B m ∈ L(L 2 (M))∩ But in our example of multiplication operators, this assumption is implied by (2.5). In the first Definition, we explain two solution concepts for problem (1.1).
; such that the equation almost surely for all t ∈ [0, T ] (see for example the proof of Proposition 3.1 in a similar situation), since the nonlinearity with α ∈ (1, 3] maps H 1 (M) to L 2 (M) by the Sobolev embedding In the following definition, we fix different notions of uniqueness. As we have seen in the previous remark, it makes sense to define uniqueness by comparing solutions in C([0, T ], L 2 (M)).

Definition 2.4.
a) The solutions of problem (1.1) are called pathwise unique in . We continue with some auxiliary results which are either well-known or due to Burq, Gérard and Tzvetkov, [11]. The first Lemma gives us an estimate for the nonlinear term in (1.1).
The following Lemma deals with a Littlewood-Paley type decomposition of L p (M) for p ∈ [2, ∞).
Then, we have
The previous Lemma indicates the importance of estimating operators of the form ϕ(h 2 ∆ g ) for h ∈ (0, 1]. In the next Lemma, we state how they act in L p -spaces and Sobolev spaces. Note that these kind of estimates are usually called Bernstein inequalities.
Fact 2.20 in [30] and the Spectral Multiplier Theorem 7.6 in [16] hence imply Since we also have For every p ∈ (1, ∞), we therefore get This completes the proof of Lemma 2.7.
In the following Lemmata, we quote the spectrally localized Strichartz estimates from [11], which are a consequence of [21]. In this paper, Keel and Tao solved the endpoint case needed for our application in the proof of Proposition 3.1.
Then, for any ϕ ∈ C ∞ c (R), there is β > 0 and C > 0 such that for h ∈ (0, 1] and any interval J of length |J| ≤ βh Proof. See [11], Proposition 2.9. The result follows from the dispersive estimate for the Schrödinger group from [11], Lemma 2.5, and an application of Keel-Tao's Theorem ( . A similar result also holds for convolutions with the Schrödinger group. Lemma 2.9. Let M be a compact riemannian manifold of dimension d ≥ 1 and p 1 , For any ϕ ∈ C ∞ c (R), there is β > 0 and C > 0 such that for h ∈ (0, 1] and any interval J of length Proof. See [11], Lemma 3.4. To prepare the next Lemma, we recall the following notation. Adapting the proof of Theorem 3.10 in [10] to the present situation, we obtain a spectrally localized stochastic Strichartz estimate for stochastic convolutions with the Schrödinger group. Then, G L 2 (Ω,L 2 (J,L 6 )) φ(h 2 ∆ g )B L 2 (Ω,L 2 (J,HS(Y,L 2 ))) .
Proof. We abbreviate and use the Burkholder-Davis-Gundy-inequality in the martingale type 2 Banach space L 2 (J, L 6 ) (see for example [7]) to estimate Writing out the definition of γ(Y, L 2 (J, L 6 )) and using ϕ( where (γ m ) m∈N is a sequence of i.i.d. N (0, 1)-Gaussians on some probability spaceΩ. By Lemma 2.8, the operator e i·∆g ϕ(h 2 ∆ g ) is bounded from L 2 (M) to L 2 (J, L 6 ). Hence, we can take it out of the sum and obtain . Finally, inserting the last estimate in (2.12) yields The proof of Lemma 2.11 is thus completed.

UNIQUENESS
In the following section, we will prove the pathwise uniqueness of solutions of (1.1). A key ingredient for this result is an L 2 t L p x -estimate for solutions for arbitrary large p with moderate growth of the bound in p. Then, there is a measurable set Ω 1 ⊂ Ω with P(Ω 1 ) = 1 such that for all ω ∈ Ω 1 , p ∈ [6, ∞) and intervals J ⊂ [0, T ], we have u(·, ω) ∈ L 2 (J; L p (M)) with u(·, ω) L 2 (J,L p ) ω 1 + (|J|p) We remark that this estimate of L p -norms is a substitute for the L ∞ -bound for solutions in the 2D-setting, see [9], and complements the inequality for p ∈ [1, 6], which we get from Sobolev's embedding and the energy estimate for martingale solutions. Before we start with the proof, we introduce an equidistant partition of the time interval. ]. Then, we have We fix ϕ,φ ∈ C ∞ c (R\{0}) withφ = 1 on supp(ϕ). In order to localize the solution u spectrally and in time, we set v I j (t) = χ I j (t)ϕ(h 2 ∆ g )u(t), j = 0, . . . , N T , and apply Itô's formula to Φ j ∈ C 1,2 ( for j = 1, . . . , N T in H −3 (M) almost surely for t ∈ I ′ j . Because of the regularity of each term (recall α ≤ 3), this identity also holds in L 2 (M). Analogously, we get in L 2 (M) almost surely for t ∈ I ′ 0 . We abbreviate We use the stochastic Strichartz estimate from Lemma 2.11, the properties of (I j ) N T j=0 and I ′ j N T j=0 and Lemma 2.7 b) to estimate Hence, there is C = C(ω) with C < ∞ almost surely such that Step 2. We fix a path ω ∈ Ω h , where Ω h is the intersection of the full measure sets from (3.2), (3.3), (3.4) and u j ∈ L ∞ (0, T ; H 1 (M)) almost surely. In the rest of the argument, we skip the dependence of ω to keep the notation simple. Let us pick those intervals J 0 , . . . , J N from the partition (I j ) N T j=0 which cover the given interval J. The associated intervals in (I ′ j ) N j=0 will be denoted by Applying the homogeneous and inhomogenous Strichartz estimates from Lemma 2.8 and 2.9 in (3.2) and in (3.3), we obtain Next, we estimate the terms on the right hand side of (3.6) and (3.7). By (3.1), Lemma 2.7 b) and Hölder's inequality, we get ) and the boundedness of the operators ϕ(h 2 ∆ g ) and µ in H 1 (M) yield We apply Lemma 2.5 with r ′ = 6 α+2 ≥ 6 5 and q = 6 and obtain the estimate where we used α ≤ 3. Together with Hölder's inequality, Lemma 2.7 b) and the boundedness of ϕ(h 2 ∆ g ), this implies . Inserting the last three estimates in (3.6) and (3.7) yields We square the estimates (3.8) and (3.9) and sum them up. Using χ J j = 1 on J j , (3.5) and Below, we will use the notations Let p ≥ 6. Then, Lemma 2.7 a) and u ∈ L ∞ (0, T ; H 1 (M)) imply Step 3. In the last step, we use (3.11) and Littlewood-Paley theory to derive the estimate stated in the Proposition. To this end, we set h k := 2 − k 2 and k 0 := min k : |J| > βh k 4 . Let us define Ω 1 := ∞ k=1 Ω h k and fix a path ω ∈ Ω 1 . We remark that we have P(Ω 1 ) = 1 by the choice of Ω h for each h ∈ (0, 1] from the previous step. In the rest of the argument, we skip the dependence of ω to keep the notation simple. Moreover, we choose ψ ∈ C ∞ c (R), Then, Lemma 2.6, the embedding ℓ 1 (N) ֒→ ℓ 2 (N) and (3.11) imply From Lemma 2.7 a) and the Sobolev embedding, we infer (3.14) We proceed with the estimate of the sums over k ≥ k 0 . The fact that we have J h k+1 ⊂ J h k for all k ∈ N, leads to Using |J h k 0 | ≤ 3 Finally, the calculation yields the boundedness of the function defined by Using the estimates (3.13) (3.14), (3.15), and (3.16) in (3.12), we get which implies the assertion. The proof of Proposition 3.1 is thus completed.
We would like to continue with some remarks on seemingly natural extensions of the previous result to higher dimensions, nonlinear noise and non-compact manifolds.

Remark 3.3.
We would like to comment on the case of higher dimensions d ≥ 4. The Strichartz-endpoint is (2, 2d d−2 ) and the use of Lemma 2.5 leads to the restriction α ≤ 1 + 2 d−2 . The corresponding estimate in (3.12) has to be replaced by Hence, the convergence of the sums requires an upper bound on p, which destroys the uniqueness proof below such that the case d ≥ 4 remains an open problem. In fact, this problem occurs since the scaling condition for Strichartz exponents, Sobolev embeddings and Bernstein inequalities are more restrictive in higher dimensions and therefore, the restriction to d = 3 is of deterministic nature.

Remark 3.4.
In the proof of Proposition 3.1, we did not need the optimal estimates for the correction term µ and the stochastic integral. In fact, it is possible to generalize the argument and show the estimate for martingale solutions of the equation 17) with nonlinear noise of power γ ∈ [1, 2). However, we do not know if this equation has a solution, since the existence theory developed in [9] only applies for γ = 1. Moreover, it is unclear how to apply these estimates in order to prove pathwise uniqueness since the arguments below rely on the linearity of the noise. Hence, the case of equation (3.17) remains another open problem.

Remark 3.5.
Let us comment on the case of possibly non-compact manifolds with bounded geometry. In the two dimensional setting, the Strichartz estimates from [3] with an additional loss of ε regularity were sufficient to prove uniqueness, see [9], Section 7. In fact, these estimates correspond to localized Strichartz estimates of the form for all ε > 0 and some C ε > 0 and β ε > 0, where we denote ψ m,a (λ) := λ m e −aλ for m ∈ N and a > 0. A continuous version of the Littlewood-Paley inequality which can substitute (2.10) is given by for each ε > 0 and p ∈ [6, 6ε −1 ) with an implicit constant which goes to infinity for ε → 0. The upper bound on p is due to the fact that the additional ε in (3.18) weakens the estimates of the critical term containing the derivative χ ′ j of the temporal cut-off and enlarges the number of summands in (3.10). As in the case of higher dimensions than d = 3, the uniqueness argument breaks down since a limit process p → ∞ is no longer possible.
So far, we only used the topological properties of the noise, i.e. Now, the Stratonovich structure and the symmetry of the operators B m for m ∈ N come into play to prove the following representation formula for the L 2 -distance of two solutions.
Note that the RHS of (3.20) only contains the terms induced by the nonlinearity. In particular, the stochastic integral vanishes, which will enable us to use the pathwise estimate from Proposition 3.1 to prove uniqueness.
Proof. We restrict ourselves to a formal argumentation. Similarly to [9], Proposition 6.5, our reasoning can be rigorously justified by a regularization procedure based on Yosida approximations R λ := λ (λ − ∆ g ) −1 for λ > 0. The function M : for v, h 1 , h 2 ∈ L 2 (M). We set w := u 1 − u 2 . Then, a formal application of the Itô formula yields almost surely for all t ∈ [0, T ]. Since ∆ g is selfadjoint, we get Re w, i∆ g w L 2 = 0. From the symmetry of B m , m ∈ N, we infer Re w, iB m w L 2 = 0 and thus, we obtain Re w(s), B 2 m w(s) Therefore, we have almost surely for all t ∈ [0, T ].
We close with the proof of our main Theorem 1.1. We prove the uniqueness by applying a strategy developed by Yudovich, [32], for the Euler equation. In the context of the NLS, it was first used by Vladimirov in [31], Ogawa and Ozawa in [26] and [27]. They looked at 2D domains and used Trudinger type inequalities to control the growth of L p -norms for p → ∞. A generalization of this argument to the stochastic case in 2D is straightforward and can be found in [19], Subsection 5.2. Following Burq, Gérard and Tzvetkov in the case without boundary, the Yudovich-strategy in combination with Strichartz estimates as an improvement of Trudinger's inequality was also applied it to the deterministic NLS on compact 3D manifolds with boundary by Blair, Smith and Sogge in [4].