The Yamada-Watanabe Theorem for mild solutions to stochastic partial differential equations

We prove the Yamada-Watanabe Theorem for semilinear stochastic partial differential equations with path-dependent coefficients. The so-called"method of the moving frame"allows us to reduce the proof to the Yamada-Watanabe Theorem for stochastic differential equations in infinite dimensions.


Introduction
The goal of the present paper is to establish the Yamada-Watanabe Theoremwhich originates from the paper [17] -for mild solutions to semilinear stochastic partial differential equations (SPDEs) dX(t) = (AX(t) + α(t, X))dt + σ(t, X)dW (t) (1.1) in the spirit of [2,12,6] with path-dependent coefficients. More precisely, denoting by H the state space of (1.1), we will prove the following result (see, e.g. [9] for the finite dimensional case): 1.1. Theorem. The SPDE (1.1) has a unique mild solution if and only if both of the following two conditions are satisfied: (1) For each probability measure µ on (H, B(H)) there exists a martingale solution (X, W ) to (1.1) such that µ is the distribution of X(0). (2) Pathwise uniqueness for (1.1) holds.
The precise conditions on A, α and σ, under which Theorem 1.1 holds true, are stated in Assumptions 2.2 and 3.1 below. So far, the following two versions of the Yamada-Watanabe Theorem in infinite dimensions are known in the literature: • For SPDEs of the type (1.1) with state-dependent coefficients α(t, X(t)) and σ(t, X(t)); see [11]. • For stochastic evolution equations in the framework of the variational approach; see [13]. We will divide the proof of Theorem 1.1 into two steps: (1) First, we show that we can reduce the proof to Hilbert space valued SDEs This is due to the "method of the moving frame", which has been presented in [5], see also [16].
(2) For Hilbert space valued SDEs (1.2) however, the Yamada-Watanabe Theorem is a consequence of [13]. The remainder of this paper is organized as follows: In Section 2 we present the general framework, in Section 3 we provide the proof of Theorem 1.1, and in Section 4 we show an example illustrating Theorem 1.1.

Framework and definitions
In this section, we prepare the required framework and definitions. The framework is similar to that in [13] and we refer to this paper for further details.
Let H be a separable Hilbert space and let (S t ) t≥0 be a C 0 -semigroup on H with infinitesimal generator A : D(A) ⊂ H → H. The path space is the space of all continuous functions from R + to H. Equipped with the metric consisting of all functions from the path space W(H) starting in zero. For t ∈ R + we denote by B t (W(H)) the σ-algebra generated by all maps W(H) → H, w → w(s) for s ∈ [0, t]. Let C(H) be the collection of all cylinder sets of the form with t 1 , . . . , t n ∈ R + and B 1 , . . . , B n ∈ B(H) for some n ∈ N, and let C (H) be the collection of all cylinder sets of the form for t 1 , . . . , t n ∈ R + and B ∈ B(H) ⊗n for some n ∈ N. Similarly, for t ∈ R + let C t (H) be the collection of all cylinder sets of the form (2.2) with t 1 , . . . , t n ∈ [0, t] and B 1 , . . . , B n ∈ B(H) for some n ∈ N, and let C t (H) be the collection of all cylinder sets of the form (2.3) for t 1 , . . . , t n ∈ [0, t] and B ∈ B(H) ⊗n for some n ∈ N.
Proof. We can argue as in the finite dimensional case, see e.g. [14, Section 2.II].
Let U be another separable Hilbert space and let L 2 (U, H) denote the space of all Hilbert-Schmidt operators from U to H equipped with the Hilbert-Schmidt norm. Let α : R + × W(H) → H and σ : R + × W(H) → L 2 (U, H) be mappings.

2.2.
Assumption. We suppose that the following conditions are satisfied: We call a filtered probability space B = (Ω, F, (F t ) t≥0 , P) satisfying the usual conditions a stochastic basis. In the sequel, we shall use the abbreviation B for a stochastic basis (Ω, F, (F t ) t≥0 , P), and the abbreviation B for another stochastic basis (Ω , F , (F t ) t≥0 , P ). For a sequence (β k ) k∈N of independent Wiener processes we call the sequence W = (β k ) k∈N a standard R ∞ -Wiener process.

Definition.
A pair (X, W ), where X is an adapted process with paths in W(H) and W is a standard R ∞ -Wiener process on a stochastic basis B is called a martingale solution to (1.1), if we have P-almost surely t 0 α(s, X) ds + t 0 σ(s, X) 2 L2(U,H) ds < ∞ for all t ≥ 0 and P-almost surely it holds In finite dimensions, a pair (X, W ) as in Definition 2.3 is called a weak solution. As in [2, Chapter 8], we use the term martingale solution in order to avoid ambiguities with the concept of a weak solution to (1.1), which means that for each ζ ∈ D(A * ) we have P-almost surely for all t ≥ 0. Sometimes, the latter concept is also called an analytically weak solution, see [12].
2.6. Remark. The stochastic integral from Definition 2.3 is defined as where J : U →Ū is a one-to-one Hilbert Schmidt operator into another Hilbert spaceŪ , andW where (e k ) k∈N denotes an orthonormal basis of U , is anŪ -valued trace class Wiener process with covariance operator Q = JJ * . Further details about this topic can be found in [12, Section 2.5]. 2.8. Definition. We say that pathwise uniqueness holds for (1.1), if for two martingale solutions (X, W ) and (X , W ) on the same stochastic basis B and with the same R ∞ -Wiener process W such that P(X(0) = X (0)) = 1 we have X = X up to indistinguishability.
2.9. Definition. LetÊ(H) be the set of maps F : H × W 0 (Ū ) → W(H) such that for every probability measure µ on (H, B(H)) there exists a map ) with respect to µ ⊗ P Q , and P Q denotes the distribution of the Q-Wiener processW on (W 0 (Ū ), B(W 0 (Ū ))). Of course, F µ is µ ⊗ P Q -almost everywhere uniquely determined. (1) For all x ∈ H and t ∈ R + the mapping denotes the completion with respect to P Q in B(W 0 (Ū )).
(2) We have up to indistinguishability 2.11. Definition. We say that the SPDE (1.1) has a unique mild solution if there exists a mapping F ∈Ê(H) such that: (1) For all x ∈ H and t ∈ R + the mapping denotes the completion with respect to P Q in B(W 0 (Ū )).
(2) For every standard R ∞ -Wiener process W on a stochastic basis B and any In this section, we shall provide the proof of Theorem 1.1. The general framework is that of Section 2. In particular, we suppose that the coefficients α and σ satisfy Assumption 2.2. As mentioned in Section 1, we shall utilize the "method of the moving frame" from [5]. For this, we require the following assumption on the semigroup (S t ) t≥0 .
3.1. Assumption. We suppose that there exist another separable Hilbert space H, a C 0 -group (U t ) t∈R on H and continuous linear operators ∈ L(H, H), π ∈ L(H, H) such is injective, we have rg(π) = H and ker(π) = rg( ) ⊥ , and the diagram Remark. According to [5,Prop. 8.7], this assumption is satisfied if the semigroup (S t ) t≥0 is pseudo-contractive (one also uses the notion quasi-contractive), that is, there is a constant ω ∈ R such that This result relies on the Szőkefalvi-Nagy theorem on unitary dilations (see e.g. [15, Thm. I.8.1], or [3, Sec. 7.2]). In the spirit of [15], the group (U t ) t∈R is called a dilation of the semigroup (S t ) t≥0 .

3.3.
Remark. The Szőkefalvi-Nagy theorem was also utilized in [8,7] in order to establish results concerning stochastic convolution integrals.
In the sequel, for some closed subspace K ⊂ H we denote by Π K the orthogonal projection on K.

3.5.
Lemma. The following statements are true: Proof. Let C ∈ C(H) be a cylinder set of the form with t 1 , . . . , t n ∈ R + and B 1 , . . . , B n ∈ B(H) for some n ∈ N. Then we have Now, our idea for the proof of Theorem 1.1 is as follows: The proof that the existence of a unique mild solution to the SPDE (1.1) implies the two conditions from Theorem 1.1 is straightforward and can be provided as in [13]. For the proof of the converse implication, we will first show that the conditions from Theorem 1.1 imply the conditions from Theorem 3.7, see Propositions 3.13 and 3.14. Then, we will apply Theorem 3.7, which gives us the existence of a unique strong solution to the SDE (1.2), and finally, we will prove that this implies the existence of a unique mild solution to the SPDE (1.1), see Proposition 3.16. For the following four results (Lemma 3.8 to Corollary 3.11), we fix a stochastic basis B = (Ω, F, (F t ) t≥0 , P).
The following auxiliary result provides us with a standard extension which we require for the proof of Proposition 3.13.

3.12.
Lemma. Let (X , W ) be a martingale solution to (1.1) on a stochastic basis B and let ν be a probability measure on (H, B(H)). Then, there exist a stochastic basis B, a martingale solution (X, W ) to (1.1) on B such that the distributions of X(0) and X (0) coincide, and a F 0 -measurable random variable η : Ω → H such that ν is the distribution of η.
Proof. We define the stochastic basis B as Proof. Let ν be a probability measure on (H, B(H)). Then the image measure µ := ν π is a probability measure on (H, B(H)). By assumption, there exists a martingale solution (X , W ) to (1.1) on a stochastic basis B such that µ is the distribution of X (0). According to Lemma 3.12, there exist a stochastic basis B, a martingale solution (X, W ) on B such that µ is the distributions of X(0), and a F 0 -measurable random variable η : Ω → H such that ν is the distribution of η. We set Y := η +
Since pathwise uniqueness for (1.1) holds, we deduce that X = X up to indistinguishability. This implies up to indistinguishability proving that pathwise uniqueness for (1.2) holds.
The following auxiliary result is required for the proof of Proposition 3.16.
3.15. Lemma. Let ν be an arbitrary probability measure on (H, B(H)). We define the image measure ν := µ on (H, B(H)). Then the mapping showing that There exists a set N ∈ B(H) ⊗ B(W 0 (Ū )) satisfying N ⊂ N and (ν ⊗ P Q )(N ) = 0. We obtain 3.16. Proposition. If the SDE (1.2) has a unique strong solution, then the SPDE (1.1) has a unique mild solution.
Proof. Suppose the SDE (1.2) has a unique mild solution. Then, there exists a mapping G ∈Ê(H) such that the three conditions from Definition 2.11 are fulfilled. In detail, the following conditions are satisfied: is a mapping such that for every probability measure ν on (H, B(H)) there exists a map • For all y ∈ H and t ∈ R + the mapping denotes the completion with respect to P Q in B(W 0 (Ū )).
We define the mapping µ⊗P Q /B(W(H))-measurable by virtue of Lemmas 3.5 and 3.15. Let us prove that F ∈Ê(H). For this purpose, let µ be an arbitrary probability measure on (H, B(H)). We define the image measure ν := µ . Then ν is a probability measure on (H, B(H)). Furthermore, we define the mapping There is a ν-nullset N ⊂ H such that for all y ∈ N c identity (3.3) is satisfied. The set −1 (N ) ⊂ H is a µ-nullset. Indeed, there is a set N ∈ B(H) satisfying N ⊂ N and ν(N ) = 0. We obtain showing that −1 (N ) ⊂ H is a µ-nullset. Let x ∈ −1 (N ) c = −1 (N c ) be arbitrary. Then we have x ∈ N c , and hence for P Q -almost all w ∈ W 0 (Ū ). Consequently, we have F ∈Ê(H). Now, we shall prove that the mapping F satisfies the three conditions from Definition 2.11. For all x ∈ H and t ∈ R + the mapping is B t (W 0 (Ū )) P Q /B t (W(H))-measurable due to Lemma 3.5. Let W be a standard R ∞ -Wiener process on a stochastic basis B, and let ξ : Ω → H be a F 0 -measurable random variable. Then the pair (Y, W ), where Y := G( ξ,W ), is a martingale solution to (1.2) with P(Y (0) = ξ) = 1. By Corollary 3.11, the pair (X, W ), where X := F (ξ,W ) = Γ(Y ), is a martingale solution to (1.1) with P(X(0) = ξ) = 1.
We deduce that up to indistinguishability Consequently, the mapping F fulfills the three conditions from Definition 2.11, proving that the SPDE (1.1) has a unique mild solution.
Now, the proof of Theorem 1.1 is a direct consequence: If the SPDE (1.1) has a unique mild solution, then arguing as in [13] shows that the two conditions from Theorem 1.1 are fulfilled. Conversely, if these two conditions are satisfied, then combining Propositions 3.13, 3.14, Theorem 3.7 and Proposition 3.16 shows that the SPDE (1.1) has a unique mild solution.

An example
In this section, we shall illustrate Theorem 1.1 and consider SPDEs of the type dX(t) = (AX(t) + B(t, X(t)) + F (t, X(t)))dt + QdW t , (4.1) which have been studied in [1], with a Hölder continuous mapping B. We fix a finite time horizon T > 0, an orthonormal basis (e n ) n∈N of H and suppose (as in [1, Section 1.1]) that the following conditions are satisfied: • A is selfadjoint, with compact resolvent, and there is a non-decreasing sequence (α n ) n∈N ⊂ (0, ∞) such that Ae n = −α n e n for all n ∈ N. x − y α .
• Q t := t 0 S s QS * s ds is a trace class operator for each t > 0.
Furthermore, in order to ensure the existence of martingale solutions, we suppose that S t is a compact operator for each t > 0. Then, as indicated in [1], strong existence holds true. Indeed, by [6, Theorem 3.14] we have the existence of martingale solutions, and by [1,Theorem 7] pathwise uniqueness holds true. Hence, according to Theorem 1.1, the SPDE (4.1) has a unique mild solution.