Yamada-Watanabe theorem for stochastic evolution equations in

The purpose of this note is to give a complete and detailed proof of the fundamental Yamada-Watanabe Theorem on infinite dimensional spaces, more precisely in the framework of the variational approach to stochastic partial differential equations.


Framework and Definitions
Let H be a separable Hilbert space, with inner product •, • H and norm • H . Let V, E be separable Banach spaces with norms • V and • E , such that V ⊂ H ⊂ E continuously and densely.For a topological space X let B(X) denote its Borel σ-algebra.By Kuratowski's theorem we have that V ∈ B(H), H ∈ B(E) and B(V ) = B(H) ∩ V , B(H) = B(E) ∩ H.
Setting x V := ∞ if x ∈ H \ V , we extend • V to a function on H.We recall that this extension is B(H)-measurable and lower semicontinuous (cf.e.g.[PR06, Exercise 4.2.3]).Hence the following path space is well-defined: Obviously, (B, ρ) is a complete separable metric space.Let B t (B) denote the σalgebra generated by all maps π s : B → H, s ∈ [0, t], where π s (w) := w(s), w ∈ B. For t 0 and w ∈ B define the stopped path w t by w t (s) = w(s ∧ t), s 0.
Below we shall use the following elementary, but useful measure theoretic facts.
Lemma 1.1.The map w → w t is B t (B)/B(B) measurable.
Proof.It suffices to show that π q (w t ) is B t (B)/B(H) measurable for q 0. But π q (w t ) = π q (w) if q t, and π q (w t ) = π t (w) otherwise.In either case, we get a B t (B)/B(H) measurable map.
Lemma 1.2.For every set A ∈ B t (B) we have 1 A (w) = 1 A (w t ).
Proof.By definition, 1 A (w) = 1 A (w t ) for sets of the form for n 1, s i t, and B i ∈ B(H).Since these sets generate B t (B), the monotone class theorem finishes the proof.
Corollary 1.3.Let (E, E) be a measurable space and f : Let (U, , 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 usual Hilbert- As usual we call (Ω, F, P, (F t )) a stochastic basis if (Ω, F, P ) is a complete probability space and (F t ) is a right continuous filtration on Ω augmented by the P -zero sets.Let β k , k ∈ N, be independent (F t )-Brownian motions on a stochastic basis (Ω, F, P, (F t )) and define the sequence Below we refer to such a process W on R ∞ as standard R ∞ -Wiener process.We fix an orthonormal basis {e k , k ∈ N} of U and consider W as a cylindrical Wiener process on U , that is, we informally have We consider the following stochastic evolution equation: Definition 1.4.A pair (X, W ), where X = (X(t)) t∈[0,∞) is an (F t )-adapted process with paths in B and W is a standard R ∞ -Wiener process on a stochastic basis (Ω, F, P, (ii) As a stochastic equation on E we have Remark 1.5.(i) By the measurability assumptions on b and σ, it follows that if X is as in Definition 1.4 then both processes b(•, X) and σ(•, X) are (F t )adapted.
(ii) We recall that by definition of the H-valued stochastic integral in (ii) we have where J is any one-to-one Hilbert-Schmidt operator from U into another Hilbert space ( Ū , Ū ) and This definition of the stochastic integral is independent of the choice of J and ( Ū , , Ū ).We refer e.g. to [PR06, Section 2.5] for details, and only mention here that for , where Q := JJ * , and that W is a Q-Wiener process on Ū .
Below we shall fix one such J and ( Ū , , Ū ) as in Remark 1.5(ii) and set σ(s, w) := σ(s, w) and for any standard R ∞ -Wiener process W we define W as in (1.2) for the fixed J. Furthermore we define equipped with the supremum norm and Borel σ-algebra B(W 0 ).For t ∈ R + let B t (W 0 ) be the σ-algebra generated by π s : W 0 → Ū , 0 ≤ s ≤ t, π s (w) := w(s).
To define strong solutions we need to introduce the following class Ê of maps: Let Ê denote the set of all maps F : H × W 0 → B such that for every probability measure µ on (H, B(H)) there exists a B(H) ⊗ B(W 0 ) denotes the completion of B(H) ⊗ B(W 0 ) with respect to µ ⊗ P Q , and P Q denotes the distribution of the Q-Wiener process on Ū on (W 0 , B(W 0 )).Of course, F µ is uniquely determined µ ⊗ P Q -a.e.. Definition 1.8.A weak solution (X, W ) to (1.1) on (Ω, F, P, where B t (W 0 ) denotes the completion with respect to P Q in B(W 0 ).Definition 1.9.Equation (1.1) is said to have a unique strong solution, if there exists F ∈ Ê satisfying the adaptiveness condition in Definition 1.8 and such that: 1.For every standard R ∞ -Wiener process on a stochastic basis (Ω, F, P, (F t )) and any F 0 /B(H)-measurable ξ : Ω → H the continuous process is (F t )-adapted and satisfies (i), (ii) in Definition 1.4, i.e. (F (ξ, W ), W ) is a weak solution to (1.1), and X(0) = ξ P -a.e.. 2. For any weak solution (X, W ) to (1.1) we have Remark 1.10.Since X(0) of a weak solution is P -independent of W , thus we have that the existence of a unique strong solution for (1.1) implies that also weak uniqueness holds.

The Main Result and its Proof
Let us now formulate the main result (see e.g.[IW81] for the finite dimensional case).
Theorem 2.1.Let σ and b be as above.Then equation (1.1) has a unique strong solution if and only if both of the following properties hold: (i) For every probability measure µ on (H, B(H)) there exists a weak solution (X, W ) of (1.1) such that µ is the distribution of X(0).(ii) Pathwise uniqueness holds for (1.1).
Proof.Suppose (1.1) has a unique strong solution.Then (ii) obviously holds.To show (i) one only has to take the probability space (W 0 , B(W 0 ), P Q ) and consider Now let us suppose that (i) and (ii) hold.The proof that then there exists a unique strong solution for (1.1) is quite technical.We structure it through a series of lemmas.
Fix a probability measure µ on (H, B(H)) and let (X, W ) with stochastic basis (Ω, F, P, (F t )) be a weak solution to (1.1) with initial distribution µ.Define a probability measure P µ on (H × B × W 0 , B(H) ⊗ B(B) ⊗ B(W 0 )), by Lemma 2.3.There exists a family K µ ((x, w), dw 1 ), x ∈ H, w ∈ W 0 , of probability measures on (B, B(B)) having the following properties: (i) For every A ∈ B(B) the map denotes the completion with respect to µ ⊗ P Q in B(H) ⊗ B(W 0 ).
Proof.Let Π : H × B × W 0 → H × W 0 be the canonical projection.Since X(0) is F 0 -measurable, hence P -independent of W , it follows that Hence by the existence result on regular conditional distributions (cf.e.g.[IW81, Corollary to Theorem 3.3 on p. 15]), the existence of the family K µ ((x, w), dw 1 ), x ∈ H, w ∈ W 0 , satisfying (i) and (ii) follows.
To prove (iii) it suffices to show that for t ∈ [0, ∞) and for all A 0 ∈ B(H), (2.1) since the system of all A ∩ A , A ∈ B t (W 0 ), A as above generates B(W 0 ).But by part (ii) above, the left-hand side of (2.1) is equal to (2.2) But 1 A ( W ) is P -independent of F t , since W is a standard R ∞ -Wiener process on (Ω, F, P, F t ), so the right-hand side of (2.2) is equal to

Define the stochastic basis
where , and define maps Then, obviously, -adapted for every x ∈ H. Furthermore, for 0 s < t, y ∈ H, and where we used Lemma 2.3(iii) in the last step.Now the assertion follows by (2.4), a monotone class argument and the same reasoning as in the proof of [PR06, Proposition 2.1.13].
Over the next few lemmas we carefully develop a pathwise definition of the stochastic integral.Some care is required as we need to consider the integral simultaneously with respect to several different stochastic bases.The definition of the stochastic integral uses the notions of predictability and stopping times, both of which depend on the underlying filtration.We start with an approximation to the integrand that is predictable with respect to the raw filtration.
Lemma 2.5.Let Z be a separable Hilbert space valued, adapted, measurable process on a stochastic basis (Ω, F, (F t ), P ).If E T 0 Z(s, ω) 2 ds < ∞, then there exist elementary predictable processes p n so that From the summability in (2.5) we see there is N 2 ∈ B(H) with µ(N 2 ) = 0 so that for x / ∈ N 2 , As in Lemma 2.4, let N 3 ∈ B(H) with µ(N 3 ) = 0 so that for x / ∈ N 3 , Π 3 is an ( Fx t )-Wiener process on ( Ω, Fx , Q x ).Now the summability in (2.5) and the isometry for stochastic integrals means that so that there is N 4 ∈ B(H) with µ(N 4 ) = 0 so that for x / ∈ N 4 , we have ∈ N .Then on the stochastic basis ( Ω, Fx , Q x ), Π 3 is a Wiener process, p k n (s, Π 1 ) are predictable processes that converge to the adapted process 1 (0,τ k (Π1)] (s)σ(s, Π 1 ) in L 2 ([0, T ] × Ω, ds × Q x ; H).So, on the one hand, while on the other hand, Adding these up as in (2.6), we find that I(Π 1 , Π 3 ) is a version of the stochastic integral T 0 σ(s, Π 1 ) dΠ 3 (s).Lemma 2.7.There exists N ∈ B(H) with µ(N ) = 0 such that for all x ∈ N , (Π 1 , Π 3 ) and (Π 2 , Π 3 ) with stochastic basis ( Ω, Fx , Q x , ( Fx t )) are weak solutions of (1.1) such that Proof.Let I T denote the function defined in (2.6).Now define a subset of B × W 0 as follows Since (X, W ) is a solution of the equation, we have So there exists N 5 ∈ B(H) with µ(N 5 ) = 0 so that for x / ∈ N 5 , we have Finally, Lemma 2.10.(X , W ) is a weak solution to (1.1) with X (0) = ξ P -a.s.. Proof.By the measurability properties of F µ (cf.Lemma 2.8) it follows that X is adapted.We have where we used Lemma 2.9 in the last step.
To see that (X , W ) is a weak solution we consider the set A ∈ B(H) ⊗ B(B) ⊗ B(W 0 ) defined in the proof of Lemma 2.7.We have to show that P ({(X (0), X , W ) ∈ A}) = 1.
To complete the proof we still have to construct F ∈ Ê and to check the adaptiveness conditions on it.Below we shall apply what we have obtained above now also to δ x replacing µ.So, for each x ∈ H we have a function F δx .Now define (2.7) F (x, w) := F δx (x, w), x ∈ H, w ∈ W 0 .