On Cherny's results in infinite dimensions: A theorem dual to Yamada-Watanabe

We prove that joint uniqueness in law and the existence of a strong solution imply pathwise uniqueness for variational solutions to stochastic partial differential equations of the form \begin{align*} \text{d}X_t=b(t,X)\text{d}t+\sigma(t,X)\text{d}W_t, \,\,\,t\geq 0, \end{align*} and show that for such equations uniqueness in law is equivalent to joint uniqueness in law. Here $W$ is a cylindrical Wiener process in a separable Hilbert space $U$ and the equation is considered in a Gelfand triple $V \subseteq H \subseteq E$, where $H$ is some separable (infinite-dimensional) Hilbert space. This generalizes the corresponding results of A. Cherny for the case of finite-dimensional equations.


Introduction
The connection between existence and uniqueness of weak and strong solutions is fundamental to the research area of stochastic differential equations. A starting point was the celebrated paper [14] by Yamada and Watanabe in 1971, in which the authors prove that weak existence and pathwise uniqueness yield the existence of a unique strong solution for finite-dimensional stochastic differential equations. Later several authors worked on a dual statement of this seminal result, i.e. on the implication Joint uniqueness in law + existence of strong solution ⇒ pathwise uniqueness. (Dual) A proof of (Dual) can be found in the works of Jacod ( [5]) and Engelbert ([4]). Unfortunately, verifying joint uniqueness in law turns out to be rather difficult in applications. In 2001, Cherny contributed a substantial improvement to this dual result by showing the equivalence of uniqueness in law and joint uniqueness in law for finite-dimensional equations in [2]. This striking result provides further structural insight into the interplay of the aforementioned notions of existence and uniqueness.
Recently the study of stochastic partial differential equations, which are necessarily infinite-dimensional equations, attracted much attention and nurtured extensive research activity in this direction. In [13], Röckner, Schmuland and Zhang extended the classical Yamada-Watanabe theorem to the framework of variational solutions for infinite-dimensional equations in Hilbert spaces. Naturally this brings up two questions, namely "Does the dual result (Dual) also hold in this infinite-dimensional framework?" and "Can Cherny's result on the equivalence of uniqueness and joint uniqueness in law be generalized to this setting?".
In this paper we give affirmative answers to both question: We prove (Dual) in the framework of the variational approach for solutions to stochastic partial differential equations of the form dX t = b(t, X)dt + σ(t, X)dW t , t ≥ 0, in a (infinite-dimensional) Gelfand triple V ⊆ H ⊆ E with a separable Hilbert space H, where W is a cylindrical Wiener process in another separable Hilbert space U . Further we prove the equivalence of uniqueness and joint uniqueness in law for deterministic initial conditions to such equations. We point out that both statements have also been stated in [12] by Qiao. For a comparison to this work, see Remark 3.6.
We stress that (Dual) and the equivalence of uniqueness and joint uniqueness in law have also been discussed for other types of equations and notions of solutions: Ondrejat provided affirmative answers to both questions in the setting of mild solutions for Banach space-valued equations in [11]. See Remark 3.5 for a more detailed comparison to his work. In [8], Kurtz deals with a more general type of stochastic equations and in particular considers (Dual) in this more general framework. However, the equivalence of uniqueness and joint uniqueness in law is not discussed in his setting.
This paper is organized as follows: In the second section we clarify notation and introduce the general framework, including the relevant notions of existence and uniqueness of solutions. The third section contains both main theorems. We present an outline of both proofs in order to render a better understanding of the detailed proofs later on. An explanation on why we have to restrict the second main theorem to deterministic initial conditions is also included. The final section contains the proofs of the main results as well as necessary preparations. Appendix A contains further preparations and, for the convenience of readers, who are not familiar with stochastic integration in detail, Appendix B reviews stochastic integration with respect to Hilbert space-valued martingales, since this will be of great importance within our proofs.

Notation
The set of all probability measures on a σ-algebra A will be denoted by M + 1 (A). Given a measure space (Ω, F , P), the σ-algebra F P denotes the completion of F with respect to P. For I = [0, T ] for T > 0 or I = R + we call (Ω, F , (F t ) t∈I , P) a stochastic basis, if F is complete with respect to P and (F t ) t∈I is a right-continuous filtration such that every zero set is contained in F 0 . In this case we denote the corresponding predictable σ-algebra by P T (if I = [0, T ]) or P ∞ (if I = R + ). We say that a process X = (X t ) t∈I on a stochastic basis is (F t )-predictable, if X is predictable and we want to stress the dependence on the underlying filtration (F t ) t∈I .

Basic Setting
Large parts of the framework presented in this subsection are as in Appendix E of [9]. Let (H, · , · H ) and (U, ·, · U ) be real separable (infinite-dimensional) Hilbert spaces with norms || · || H and || · || U , respectively. Further let V and E be real separable Banach spaces with norms ||·|| V , ||·|| E , respectively, such that V ⊆ H ⊆ E continuously and densely. Then Kuratowski's theorem [7, p.487 is B(H)-measurable and lower semicontinuous on H. Thus the path space We define a filtration on B by B t (B) := σ(π s |0 ≤ s ≤ t) for any t ≥ 0. Further (B + t (B)) t≥0 denotes the corresponding right-continuous filtration. Here π t : B → H is the canonical projection, i.e. π t (ω) = ω(t) for ω ∈ B. Note that (B, ρ) is a complete separable metric space, with metric ρ defined through We denote the Borel σ-algebra of (B, ρ) by B(B).

The stochastic differential equation under investigation
We consider stochastic differential equations of the form which is a formal notation for the integral equation X t = X 0 + t 0 b(s, X)ds + t 0 σ(s, X)dW s , t ≥ 0, where the first integral is a pathwise E-valued Bochner-integral and the second one is an H-valued stochastic Itô-integral. We assume that b : R + × B → E , σ : R + × B → L 2 (U, H) and W = (W t ) t≥0 fulfill the following properties.
where (e k ) k∈N is an orthonormal basis of U and (β k ) k∈N is a family of independent real-valued (F t )-Brownian motions on Ω. We also write W = (β k ) k∈N and call W a standard R ∞ -Wiener process.
The stochastic integral in (1) is defined through t 0 σ(s, X)dW s := t 0 σ(s, X) • J −1 dW s . Here J : U →Ū is a one-to-one Hilbert-Schmidt-map with values in a separable Hilbert space (Ū , ·, · Ū ) and W t := ∞ k=1 β k (t)Je k , t ≥ 0, is the cylindrical Wiener process associated to W . The orthonormal basis (e k ) k∈N is the same as in Assumption 1 above, which we fix from now on. Such J andŪ always exist and the definition of the stochastic integral does not depend on the choice of J orŪ . FurtherW is aQ-Wiener process withQ := JJ * ∈ L + 1 (Ū ). We fix such J,Ū andQ from now on. For technical details about stochastic integration with respect to cylindrical Wiener processes we refer to [9,Section 2.5.].

Strong, weak solutions and notions of uniqueness
We now present the relevant notions of solutions and uniqueness for our considerations and clarify the relations between them.
is an (F t )adapted process with paths in B and W is a standard (F t )-R ∞ -Wiener process on some stochastic basis (Ω, F , (F t ) t≥0 , P) such that the following holds true: s. as an equation on E. We call X a solution process of Eq. (1) or simply solution. Note that such X is F /B(B)-measurable. Definition 2.2. (i) Weak uniqueness (also uniqueness in law ) holds for Eq. (1), if for any two solutions (X, W 1 ) and (Y, W 2 ) on (possibly different) stochastic bases (Ω, F , (F t ) t≥0 , P) and implies P • X −1 = P ′ • Y −1 as measures on (B, B(B)).
(ii) Weak uniqueness given µ ∈ M + 1 (B(H)) holds, if the implication in (i) is at least valid for all weak solutions (X, W 1 ), (Y, W 2 ) with initial distribution µ.
(iii) Eq. (1) has joint uniqueness in law (also joint weak uniqueness), if in the setting of (i) (2) implies . The definition of joint uniqueness in law given µ ∈ M + 1 (B(H)) is analogue to (ii).
(iv) δ-weak uniqueness and δ-joint weak uniqueness hold, if the respective implications in (i) and (iii) hold at least when (2) is restricted to for every x ∈ H, i.e. to arbitrary deterministic initial conditions. δ x denotes the Dirac-measure in x.
(ii) For µ ∈ M + 1 (B(H)), pathwise uniqueness given µ means that the implication in (i) holds at least for all weak solutions as in (i), which additionally satisfy P • X −1 0 = µ = P • Y −1 0 . (iii) δ-pathwise uniqueness holds, if the implication in (i) holds at least for all solutions X and Y with X 0 = x = Y 0 for any x ∈ H.
In order to define the notion of a strong solution, letÊ denote the set of all maps F : holds. PQ denotes the distribution of theQ-Wiener processW on (W 0 , B(W 0 )). Obviously each F µ is uniquely determined up to a µ ⊗ PQ-zero set.
-measurable for every t ≥ 0 and for every standard (F t )-R ∞ -Wiener process W on any stochastic basis (Ω, F , (F t ) t≥0 , P) and any F 0 /B(H)-measurable map ξ : Ω → H, the B-valued process X := F P•ξ −1 (ξ,W ) is such that (X, W ) is a weak solution to Eq. (1) with X 0 = ξ P-a.s. We will conventionally call F the strong solution.

Main Results
We present the two main theorems of this paper. We give outlines of their proofs and point out why we have to restrict the second theorem to deterministic initial conditions. The sketch of a simple proof for a very special case of the second theorem is included as well in order to demonstrate the idea we follow for the general version. We assume the framework of the previous section to be in force.
Theorem 3.1. Consider the stochastic evolution equation where we assume that b, σ and W fulfill Assumption 1. If this equation has a strong solution and joint uniqueness in law given µ holds for some µ ∈ M + 1 (B(H)), then pathwise uniqueness given µ holds as well. In particular, the existence of a strong solution and joint uniqueness in law imply pathwise uniqueness.
Theorem 3.2. Consider the stochastic evolution equation where we assume that b, σ and W fulfill Assumption 1. For any x ∈ H, weak uniqueness given δ x is equivalent to joint uniqueness in law given δ x . In particular, δ-uniqueness in law is equivalent to δ-joint uniqueness in law.
In particular, we obtain the following corollary, which we interpret as a dual statement to the Yamada-Watanabe theorem. Corollary 3.3. Assume b, σ and W fulfill Assumption 1. Then for the stochastic differential equation above the existence of a strong solution and δ-weak uniqueness imply δ-pathwise uniqueness.

Scheme of proof of Theorem 3.1:
The proof is similar to the one presented by Cherny for the finite-dimensional case in [2,Thm. 3.2]. Assume there exists a strong solution F and joint uniqueness in law given µ holds for some µ ∈ M + 1 (B(H)). We want to prove that every weak solution is given by the strong solution F .
The main idea is to consider the regular conditional distribution of Z with respect to a suitable sub-σ-algebra of F , namely the P-completion of σ(ξ 0 ,W ), which in the proof will be called GW 0 . We will prove that the regular conditional distribution of X with respect to the same σ-algebra coincides with that of Z. This step will heavily rely on the assumption on joint uniqueness in law given µ. From here the definition of regular conditional distributions will imply E[g(X)|GW 0 ] = g(Z) for any R-valued, bounded, measurable g. By joint uniqueness in law given µ, we will easily derive g(X) = g(Z) for all g as above and from there the result is immediate. The "in particular"-statement of the theorem then follows directly, because joint uniqueness in law is, by definition, equivalent to joint uniqueness in law given µ for all µ ∈ M + 1 (B(H).

Scheme of proof of Theorem 3.2:
First of all we would like to point out that the proof would be straightforward, if we assumed the operator σ(t, y) to be one-to-one for all (t, y) ∈ [0, +∞[×B. Indeed, in this case σ −1 (t, y) is welldefined on Im σ(t, y) and for a weak solution (X, W ) we can, setting N t : Here we used Proposition B.21 (ii) from Appendix B to obtain the well-definedness of the second (hence also the first) term and to deduce the equality of the second and third integral. Thus we have expressed the Wiener process as a measurable functional of the solution X, which yields the desired statement. Although this simple reasoning does not work in the general case we consider, one will recognize the same idea in our proof below. For the general case we basically follow the ideas of Theorem 3.1. in [2] and Theorem 1.6 in [12]. The majority of techniques used for the finite-dimensional case has to be modified for our infinite-dimensional variational approach.
Fix a deterministic initial condition x ∈ H for which uniqueness in law given δ x holds and for which Eq. (3) has at least one weak solution. We prove that P • (X,W ) −1 is uniquely determined by P • X −1 for every weak solution (X,W ) with X 0 = x P-a.s. Since we assume uniqueness in law given δ x , this implies the desired statement. Roughly speaking, we will express the Wiener processW as a functional of X and a process independent of X. We will arrange the proof in the following steps: (i) Let (X, W ) be a weak solution on a stochastic basis (Ω, F , (F t ) t≥0 , P) such that X 0 = x P-a.s., let (Ω ′ , F ′ , (F ′ t ) t≥0 , P ′ ) be a second stochastic basis and W 1 , W 2 two independent R ∞ -Wiener processes on it. Consider the product spaceΩ := Ω × Ω ′ withP := P ⊗ P ′ and the obvious σalgebra and filtration such that we obtain a stochastic basis. We define the processesX,W ,W 1 andW 2 on this product space in an obvious way via projections and check that (X,W ) is also a weak solution subject to the initial condition x.
(ii) For a linear subspace V ⊆ U , let pr V denote the orthogonal projection onto V . We define the processes φ and ψ : R + × B → L(U ) via φ(t, y) := pr kerσ(t,y) ⊥ and ψ(t, y) := pr kerσ(t,y) , which we will use to split up the integralW t = t 0 J • id U • J −1 dW s later on. We further introduce the processes Jψ(s,X) dW s and verify that these are independent Wiener processes onΩ, for which we will need a Hilbert space version of Lévy's characterization of Brownian motion. Next we show that the pair (X, V 1 ) is a weak solution to Eq. (3).
(iii) In this crucial step we prove the independence ofX andV 2 . We will heavily use Lemma 4.4 as well as the assumption on uniqueness in law given δ x . More precisely, we will even show thatX is independent ofF 0 ∨ σ(V 2 t |t ≥ 0).

Due to
Step (iii) we know thatX is independent ofV 2 . The first summand is a measurable functional ofX. This will imply the result. The "in particular"-statement of the theorem is then obvious, because δ-(joint) uniqueness in law is by definition equivalent to (joint) uniqueness in law given δ x for all x ∈ H.
Remark 3.4. With our techniques, Theorem 3.2 cannot be generalized to non-deterministic initial conditions. Why is this so? Within the proof of Theorem 3.2 we crucially use Lemma 4.4, as outlined in Step (iii) above. The main point is to obtain -using the notation of Lemma 4.4 -that Hence a necessary and sufficient condition is that any weak solution X fulfills Due to the separability of H, this is equivalent to P • X −1 0 = δ x for some x ∈ H.
Remark 3.5. In [11], M. Ondrejat considers, among other statements, the assertions of both Theorem 3.1 and Theorem 3.2 in the setting of mild solutions to Banach space-valued stochastic differential equations (c.f. Theorem 1 and Theorem 4 in [11], respectively). To retrieve the type of equations we consider, X needs to be a separable Hilbert space. Further, necessarily X = X 1 in order to choose S t = id X , which is requisite to obtain our type of equations. This shows that the situation in [11] does not contain our approach via a generalized Gelfand triple. Above that one notices that the drift and diffusion term of his type of equations do not depend on entire solutions paths, but only on its current time value.
Remark 3.6. In [12] H. Qiao states both main theorems of this paper for the same type of equations and within the same framework. Two rather short proofs are given, which mostly follow the same arguments as in Cherny's proofs in [2] for the finite-dimensional setting. In doing so, central technical issues arising from the infinite-dimensional framework are not properly adjusted to the proof of Theorem 1.6. in [12]. In particular this includes (assuming the notation of [12]) the proof of the independence of V 1 and V 2 and the calculation of the covariation of JV i , i ∈ {1, 2} (note that the reference Proposition 3.13. given for this argument does not apply to the situation on p.372 in [12], because the stochastic integrals φ(·,X).W and ψ(·,X).W 1 are not necessarily independent processes). Further the well-definedness of V 1 and V 2 is not discussed and there is no justification for the computations of stochastic integrals on p.373. The final conclusion of the proof is rather imprecise. Furthermore, the important technical preparations in [12], namely Lemma 2.2. and Lemma 2.3., seem to rely heavily on arguments presented in [13] (c.f. Lemmas 2.4, 2.5 and 2.6 and the arguments inbetween). However, the situation there is, albeit quite similar in nature, technically a different one. Hence we believe it is valuable to present detailed proofs for these technical preparations as well for the main theorem.
Concerning the proof of Theorem 1.7. in [12], note that the situation considered there is less general then our setting in terms of the definition of a strong solution. Below we give a proof considering this more general notion of a strong solution, which is also more precise and detailed.

Proofs of the main results
The first subsection contains the main technical preparations for the proofs of our main results, which are presented in the second subsection.

Preparations
As before, let H and U be separable, infinite-dimensional Hilbert spaces. We start by recalling the definition and basic properties of regular conditional distributions, since these will be a key tool within the main proofs below.
The statements of the following remark are well-known results. Thus we omit their proofs.
with exception set possibly depending on A.
(ii) For X, G and E as above a unique regular conditional distribution exists.
denotes expectation with respect to Q ω for fixed ω ∈ Ω.
For the next two lemmatas we fix the following framework. Let (X, W ) be a weak solution of Eq. (1) on a stochastic basis (Ω, F , (F t ) t≥0 , P) with initial condition X 0 = x P-a.s. for some x ∈ H and let (P ω ) ω∈Ω be the regular conditional distribution of the random variable (X,W ) : Ω → B × W 0 , B(B) ⊗ B(W 0 ) with respect to F 0 (by the remark above such a r.c.d. exists, because B × W 0 is a complete separable metric space when equipped with the product metric of ρ and ζ as introduced in Subsection 1.2). For ω ∈ Ω define a stochastic basis through :Ω → W 0 denote the canonical projections on the first and second variable, respectively. Note that, as pointed out in Remark 3.6 above, the following two statements are in spirit of Lemmata 2.4, 2.5 and 2.6 in [13] and also that Lemma 4.4 below is reminiscent to Lemma 3.3. in [2].
Proof. Since Π 2 : B × W 0 → W 0 and due to the definition of W 0 , the paths of Π 2 trivially start in zero and are continuous. The (F ω t )-adaptedness of (Π 2 (t)) t≥0 is obvious for every ω ∈ Ω, so it remains to verify that there exists N 0 ∈ F with P(N 0 ) = 0 such that for ω ∈ N c 0 we have for all s, t ∈ Q with 0 ≤ s < t, because then the assertion follows by an approximation of arbitrary , A 0 ∈ F arbitrary and obtain for y ∈Ū : Above we used Remark 4.2 (iv) for the first and last equality and the independence ofW t −W s and F s in the second equation. By varying A 0 in F , we obtain P-a.s.: The last equality follows by the independence ofW t −W s from F 0 and again Remark 4.2 (iv). In particular, choosing A 1 = B and A 2 = W 0 , we obtain for all y in a countable, dense subset ofŪ : which by the uniqueness of the Fourier-transform implies that for P-a.a. ω ∈ Ω Further note that the exception set in (5) can, for fixed 0 ≤ s < t, be chosen independently of A 1 , A 2 , because both B s (B) and B s (W 0 ) are countably generated. Then the usual monotone class argument, together with Lemma A.1, shows that Π 2 (t) − Π 2 (s) is P ω -independent ofF ω s for P-a.a. ω ∈ Ω for all s, t as above.
The following statement will be crucial for the proof of Theorem 3.2. A similar result for the finite-dimensional setting is a main tool for Cherniy's result in [2] (c.f. Lemma 3.3. therein). Due to its importance for our main proof below, we decided to give a detailed proof of this lemma for our infinite-dimensional framework. BelowΠ 2 denotes the formal standard R ∞ -Wiener process associated to Π 2 .
The proof is split into two steps. We work with a sequence of elementary processes (p n ) n∈N , which approximates σ in L 2 (L 2 (U, H); P X ), because only for elementary integrands we have a pathwise definition of the stochastic integral. This pathwise definition is necessary in order to allow us to "put ω in the integrand as well as in the integrator" and thereby "put Π 1 and Π 2 in the right places". This step becomes apparent in (12) and in the definition of the setB t .
Proof. Of course Π 1 is B-valued and (F ω t )-adapted for every ω ∈ Ω. By the previous lemma, Π 2 is an (F ω t )-Q-Wiener process on (Ω,F ω , (F ω t ) t≥0 , P ω ) for P-a.a. ω ∈ Ω. LetΠ 2 be the associated standard R ∞ -Wiener process. Concerning integrability, fix t ≥ 0 and note that for P-a.a. ω ∈ Ω and all zero sets can obviously be chosen independently of t. Hence we only need to verify the following: For T > 0 there exists a P-zero-set N 2 ∈ F such that for all ω ∈ N c 2 : We prove assertion (I) in two steps.
(i) Here we assume where P X denotes the distribution of X : .., j n − 1}, has finite image and 0 = t n 0 < ... < t n jn = T is a finite partition of [0, T ]. We immediately observe and that p n (·, X) is still elementary and (F t )-predictable. Thus p n (·, X)•J −1 ∈ Λ 2 T (W ,Ū , H, P T ) and by the isometry stated in Proposition B.16, (7) yields Since conditional expectation is an L p -contraction for p ≥ 1, we obtain Hence, by (7), there exists a subsequence (n k ) k∈N such that Applying the isometry for stochastic integrals once more (this time for the Wiener processΠ 2 and the admissible integrands p n k (·, Π 1 ) and σ(s, Π 1 )) we conclude by (9): For every t ∈ [0, T ], for P-a.a. ω ∈ Ω we have Now we consider (8) only along the same subsequence (n k ) k∈N . Then there is a further subsequence (n k l ) l∈N , for which for every t ∈ P-a.s. Note that since (p n ) n∈N is a sequence of elementary processes, the stochastic integral on the left hand side in (11) is defined pathwise, i.e.  (10) implies P {(X,W ) ∈B c t } = 0. For every t ∈ [0, T ], we conclude 0 = P (X,W ) ∈B c t = Ω P ω (B c t ) P(dω), which gives P ω (B c t ) = 0 P-a.s. and thus in turn for P-a.a. ω ∈ Ω: But now (10) especially holds along the same subsequence (n k l ) l∈N . Choosing a further subsequence (possibly depending on ω and t) for which the convergence in (10) holds P ω -a.s., we conclude together with (12): By the continuity in E of all terms, the zero set N t can be chosen independently of t ∈ [0, T ]. Hence this case is settled.
which, by Fubini's theorem, is an (B + t (B))-stopping time for every k ∈ N. We continue with the following observations.
(a) For every k ∈ N and T > 0, (6) is fulfilled when one replaces σ by 1 ]0,τ T k ] σ and 1 ]0,τ T k ] σ : Hence, as in the previous step, we find elementary, (B + t (B))-predictable functions (q T,k n ) n∈N with and therefore, by the isometry for stochastic integrals, also E sup As in (9), we find a subsequence (n m ) m∈N such that Similarly to (10) we obtain for P-a.a. ω ∈ Ω : in L 2 (Ω, P ω ; H). Considering (14) along the same subsequence (n m ) m∈N yields a further subsequence (n m l ) l∈N with Proceeding along the same steps as in part (i) up to (12) with the necessary technical adjustments, we arrive at for P-a.a. ω ∈ Ω. Comparing with (15), we observe P ω -a.s.
for P-a.a. ω ∈ Ω. Now consider (16) for all k ∈ N simultaneously and pass to the limit of τ T k (Π 1 ) for k → ∞, which, as we stated above, is P ω -a.s. equal to T for P-a.a. ω ∈ Ω. By the continuity of all terms involved, for P-a.a. ω ∈ Ω Repeating this procedure for every T > 0 and using the continuity of both sides of the equation as E-valued processes, we obtain the statement.
Finally consider (II). Due to X 0 ≡ x, we have for each A ∈ B(H): and thereby P ω {Π 1 (0) ∈ A} = P(X 0 ∈ A) for P-a.a. ω ∈ Ω. Since H is a separable Hilbert space, we can choose a ∩-stable, countable generator of B(H). Then the above equality holds for all elements A of this generating set outside one common P-zero set and from there we conclude for P-a.a. ω ∈ Ω as measures on B(H), which finishes the proof.
Throughout the proof of our main results we will work with stochastic integrals, which involve certain projection-valued operators as integrands. The next lemma states that these integrals are well-defined.
(ii) By (i) and because J ∈ L(U,Ū ), both J • φ(·, X) and J • ψ(·, X) are strongly measurable, (B t (B))-adapted and L 2 (U,Ū )-valued. Now fix (t, y) ∈ R + × B. For A ∈ L 2 (U,Ū ) the value 2 is independent of the orthonormal basis {f k } k∈N . Hence we may choose {f k } k∈N such that either f k ∈ ker σ(t, y) or f k ∈ ker σ(t, y) ⊥ for every k ∈ N. Then we obtain which completes the proof of (ii), because the ψ-integral can be treated similarly.
Our next goal is to prove that the quadratic cross variation of two stochastic integrals is additive, if the integrators are independent Wiener processes (c.f. (18) below). We will need this result along the proof of our second main theorem. We start with a technical lemma. Its proof is postponed to the appendix.

Now we can straight forward prove the desired result:
Proposition 4.7. Let φ k ∈ Λ 2 T (W k , U, H, P T ) for every T > 0 for k ∈ {1, 2} and W 1 , W 2 as above. Then we have P-a.s.: for every t ≥ 0.
Proof. Let (f k ) k∈N be an orthonormal basis of H. Lemma 4.6 and the fact that bounded linear operators interchange with stochastic integrals imply for every i, j ∈ N : because by assumption on φ k , the integrands φ k (·), f i H obviously fulfill the assumption of the previous lemma for every k ∈ {1, 2} and i ∈ N. Hence the assertion follows by Corollary B.7.
Finally we present a definition, which will be useful within the proof of Theorem 3.2.
Definition 4.8. Let H be a separable Hilbert space with inner product ·, · H . The Hilbert space (H ⊕ H, ·, · H⊕H ) is defined as the Cartesian product H×H with the inner product (h 1 , h 2 ), (h 3 , h 4 ) H⊕H := h 1 , h 3 H + h 2 , h 4 H . When no confusion is possible, we abbreviate ·, · H⊕H by ·, · ⊕ . Remark 4.9. It is obvious that the Hilbert space (H ⊕H, ·, · H⊕H ) is separable and that B(H ⊕H) = B(H) ⊗ B(H). The latter holds, because the metric induced by ·, · ⊕ induces the product topology on H ⊕ H.

Proofs of the main results
Now we give proofs for the two main results of this paper.
for every weak solution X with respect to the same data. To do so, let F , Z and X be as above and set GW 0 := σ(ξ 0 ,W ) P . As before,W denotes theŪ -valued (F t )-Q-Wiener process associated to W . We make the following observations: is F 0 -measurable and hence P-independent ofW , we obtain P • (ξ 0 ,W ) −1 = µ ⊗ PQ and thereby the claim follows by the GW 0 /B(H) ⊗ B(W 0 ) µ⊗PQ -measurability of (ξ 0 ,W ) : Ω → H × W 0 . Here PQ denotes the measure P •W −1 on B(W 0 ).

Since (B, B(B))
is Polish there exists a unique regular conditional distribution of Z : Ω → B with respect to GW 0 , which we denote by (Q Z ω ) ω∈Ω . Since Z is GW 0 -measurable, Remark 4.2 implies Q Z ω = δ Z(w) P-a.s. As we assume joint uniqueness in law given µ and we have X 0 = ξ 0 = Z 0 P-a.s. and P • ξ −1 By the same arguments as above there exists a unique regular conditional distribution of X : Ω → B with respect to GW 0 , which we denote by (Q for each bounded, measurable g : B → R. Now we can finally verify (19): Fix an orthonormal basis {f i } i∈N of H and set σ j : H → R, σ j = ·, f j H . For q ∈ Q + and j, n ∈ N, define g j,n q : B → R through g j,n q (y) := σ j (π q (y)) ∧ n ∨ −n, y ∈ B and note that these functions are clearly bounded and B(B)/B(R)-measurable. As above, π q : B → H denotes the canonical projection from B to H at time q. We have lim n→∞ g j,n q (X(ω)) = X q (w), f j H for every ω ∈ Ω. Applying (22) to g j,n q for every q, j, n, we obtain X q = Z q for all q ∈ Q + P-a.s. and the path-continuity of X and Z in H completes the proof. Generalized assumption : For every triple (Ω, F , (F t ) t≥0 , P), W, ξ 0 for which at least one weak solution X exists (i.e. the pair (X, W ) is a weak solution on this stochastic basis with X 0 = ξ 0 P-a.s.), there also exists a solution Z : Ω → B subject to this triple, which is GW 0 /B(B)-measurable.
We now turn to the proof of Theorem 3.1. We will heavily need several properties and computation rules of stochastic integrals with respect to arbitrary square-integrable, continuous martingales. These properties are well-known to experts on stochastic integration in infinite dimensions. Nevertheless, for the convenience of the reader, we review the construction and properties of such stochastic integrals in Appendix B.
Proof of Theorem 3.2: Fix x ∈ H and assume uniqueness in law given δ x holds. We prove the following: For any weak solution (X, W ) to on a stochastic basis (Ω, F , (F t ) t≥0 , P), the joint distribution P • (X,W ) −1 is uniquely determined by P • X −1 . Here and for the rest of the proof, for a R ∞ -Wiener process W we denote byW theŪ -valued Q-Wiener process associated to W . As we pointed out in the recap on cylindrical Wiener processes in the second section, we haveQ = JJ * . Let us fix a weak solution (X, W ) to Eq. (23).
is an independent family of R-valued (F ′ t )-Brownian motions on Ω ′ and (e k ) k∈N is the orthonormal basis of U we fixed in Subsection 1.2. The R ∞ -Wiener processes associated toW 1 andW 2 , i.e. the families (β 1 k ) k∈N and (β 2 k ) k∈N , will be denoted by W 1 and W 2 , respectively.
The first equality is due to the convergence on the right-hand side in (31) in L 2 Ω, F , P; C([0, T ], R) and due to the uniqueness of the covariation process of continuous martingales. For the third equality, consider (32) along a subsequence (N l ) l∈N for which Jφ(s,X)e k , a Ū · Jψ(s,X)e k , b Ū ds P-a.s., since we can interchange the limit with the integral, because for fixed ω ∈ Ω the function t → ||φ(t,X(ω))J * a|| U · ||ψ(t,X(ω))J * b|| U is, by Cauchy-Schwarz-inequality, a dominating L 1 ([0, T ], dt; R)-function of the sequence Jφ(s,X(ω))e k , a Ū · Jψ(s,X(ω))e k , b Ū l∈N , so that Lebesgue's dominated convergence theorem applies. The last expression equals zero because of (29). Hence (V 1 ,V 2 ) is an (F t )-Q ⊕ -Wiener process. Consequently we have the following expressionP-a.s. independently of t ≥ 0: wheref i is defined throughf i := (f i+1 2 , 0) for i ∈ 2N 0 + 1 andf i := (0, f i 2 ) for i ∈ 2N and the series converges in L 2 (Ω, F , P; C([0, T ],Ū ⊕Ū )) for every T > 0. Here {f n |n ∈ N} denotes an orthonormal basis ofŪ consisting of eigenvectors ofQ. It is obvious that {f n |n ∈ N} is an orthonormal basis ofŪ ⊕Ū consisting of eigenvectors ofQ ⊕ . Furtherλ n is the corresponding eigenvalue off n and {β ′ n |n ∈ N} is an independent family of real-valued (F t )-Brownian motions onΩ. From the definition off n and (33) we immediately obtain P-a.s.: Since the σ-algebras σ(β ′ n (t)|t ≥ 0, n ∈ 2N 0 + 1) and σ(β ′ n (t)|t ≥ 0, n ∈ 2N) areP-independent and clearly σ( , we have proved the independence of (V 1 t ) t≥0 and (V 2 t ) t≥0 . In the sequel we will use the notation V i for the formal R ∞ -Wiener process associated toV i . The next step is to prove that (X, V 1 ) is a weak solution to (23) on (Ω,F , (F t ) t≥0 ,P) (in fact even with respect to the bigger filtration (G + t ) t≥0 as we shall see below) and thatX andV 2 areP-independent. Finally, let us again apply Proposition B.21 (i) and the two chains of equations above to obtain the following: which holdsP-a.s. for each t ≥ 0 with zero set independent of t ≥ 0.
2.X andV 2 are independent onΩ with respect toP : We first show that (X,V 1 ) remains a weak solution when replacing the filtration (F t ) t≥0 by (G + t ) t≥0 , which is the right-continuous filtration associated to G t :=F t ∨ σ(V 2 s |s ≥ 0), t ≥ 0. By Lemma A.1 we only need to show thatV 1 is a (G t )-Q-Wiener process. Obviously Since sets of the form A t ∩ D for A t and D as above form a ∩-stable generator of G t , we obtain the independence of σ(V 1 s −V 1 t |s ≥ t) and G t , which yields thatV 1 is a (G t )-Q-Wiener process onΩ.
The "in particular"-assertion of the statement is now a trivial consequence of what we just proved. and clearly 1 ]0,τ ] φ k ∈ Λ 2 T (W k , U, R, P T ) for k ∈ {1, 2}. By the construction of the stochastic integral (see Proposition B.16), it is sufficient to prove for all Φ k ∈ E T (U, R). To this end let 0 = t 0 < ... < t N = T be a finite partition of [0, T ] and set Φ k := Recall that each B k l is a map from Ω to L(U, R), which takes finitely many values {β k l1 , ..., β k lK k } and is F t l -measurable. We may assume that Φ 1 and Φ 2 have the same partition. Then For the third equality we used that W 1 and W 2 are (F t )-Wiener processes and the F t l ∨tm -measurability of 1 B which follows from the independence of W 1 and W 2 . This gives (38).
Finally, we elaborate the conclusion of the proof of Theorem 3.2 in detail. Consider the situation of the final step of the proof.
Conclusion of proof of Theorem 3.2: Consider a bounded B(B) ⊗ B(W 0 )-measurable function F : B × W 0 → R, which is continuous with respect to the topology of pointwise convergence in W 0 . We show that for every such F the integral Ω F (X,W )dP(ω) only depends on the distribution ofX underP. Indeed, we can calculate as follows: .
All limits are understood in the sense of pointwise convergence in t ≥ 0 for fixedω ∈Ω taken out of a set of fullP-measure. For an orthonormal basis (ū l ) l∈N ofŪ , the second equality follows directly from Proposition 2.4.5. in [9] andV 2 = ∞ k=1 Je k β kP -a.s. (where (e k ) k∈N denotes the fixed orthonormal basis of U and (β k ) k∈N is a family of independent, real-valued (F t )-Brownian motions onΩ). The third equality holds due to [6, Remark 2.8.7] for suitable (N j ) j∈N and an increasing sequence (t i ) i∈N ⊆ R + . All limits can be interchanged with F due to the continuity of F in the aforementioned sense and can be taken out of the integral, since F is bounded. For (n, m, j) ∈ N 3 we continue, using Proposition 2.2.2. of [9] for the second equality (note β k (t) = V 2 (t), Je k Ū and recall the independence ofX and V 2 ): Finally, we rewrite the last term on the right-hand side as Jψ(t i , y)e k ,ū l Ū w ti+1∧· , Je k Ū − w ti∧· , Je k Ū ū l dPV 2 (w)dPX (y).
SinceV 2 is aQ-Wiener process, each F (n, m, j) only depends on the distribution ofX underP, which yields this also for Ω F (y, w)dP • (X,W ) −1 (y, w). Since the set of all integrals over such F determines a measure on B(B) ⊗ B(W 0 ) uniquely, the joint distribution ofX andW underP only depends oñ P •X −1 .

B The stochastic integral for Hilbert space-valued martingales
In this section we briefly recall the construction of the stochastic integral with respect to continuous, square-integrable Hilbert space-valued martingales as integrators and state its most important properties. Most parts of this section are standard and can be found in Section 3.4 of [3] and Section 14 in [10].
Let (Ω, F , (F t ) t∈[0,T ] , P) be a stochastic basis and U a separable Hilbert space with an orthonormal basis {e n } n∈N . We introduce the Banach space By the maximal inequality, || · || M 2 T is equivalent to the L 2 (Ω, F , P; L ∞ ([0, T ]; U ))-norm on M 2 c (T ; U ).
The quadratic variation of a Hilbert space-valued, square-integrable continuous martingale It is well-known that for M ∈ M 2 c (T ; U ) (with M 0 = 0) there exists a unique real-valued, increasing, is an R-valued (F t )-martingale for all a, b ∈ U.
Next we define the notion of the quadratic cross variation for two Hilbert space-valued martingales and draw a connection to the quadratic variation reminiscent to the real-valued case.
Proof. The claim follows immediately by Definitions B.1, B.5 and the bilinearity of the cross variation.
Corollary B.7. (i) Let M, N ∈ M 2 c (T ; U ) be such that the real-valued continuous martingales M, e i U and N, e j U have covariation zero for every i, j ∈ N. Then we have for every t ∈ [0, T ] (ii) In particular (39) holds, if M and N are independent.
Finally, we state the Hilbert space-version of Lévy's characterization of Brownian motion, which is used in the proof of Theorem 3.2.

The construction of the stochastic integral
We continue with the construction of the stochastic integral. The following construction and results are standard. One starts with the construction of stochastic integral with respect to elementary integrands and then extends this definition through a suitable isometry. In the sequel H denotes another separable, (infinite-dimensional) Hilbert space. (ii) For every u ∈ U the process X • Q

Properties of the stochastic integral
The following proposition and its proof can be found in Section 4.3 of [3].
Proposition B.18. Let Q ∈ L + 1 (U ), W ∈ M 2 c (T ; U ) be a (F t )-Q-Wiener process and A ∈ Λ 2 T (W, U, H, P). Then It is well known that H 1 .(H 2 .M ) = (H 1 · H 2 ).M holds in the case of finite-dimensional stochastic integration. We use the Hilbert space-analogue of this result, stated in Proposition B.20 below, multiple times within our main proofs. This proposition and Lemma B.19 are taken from [1] (c.f. Lemma 3.6. and Theorem 3.7. therein). We also need two slight generalizations of this result, which we both state and prove in Proposition B.21 at the end of this appendix. Let G denote another separable, infinite-dimensional Hilbert space. Using Lemma B.19 and Proposition B.20 we obtain B ∈ Λ T (A.W, H, G, P T ) and that (τ n ) n∈N is also a proper localizing sequence for B. Therefore For n → +∞, the right-hand side of (41) clearly converges P-a.s. to Finally, we mention that the entire construction and all properties presented in this section immediately carry over to the case T = +∞. We would like to stress, however, that these extended stochastic integrals on Ω × R + are in general only continuous local martingales.