Yamada-watanabe Theorem for Stochastic Evolution Equations in Infinite Dimensions

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. 1. 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: B := { w ∈ C(R+;H) ∣∣∣∣ ∫ T 0 ‖w(t)‖V dt <∞ for all T ∈ [0,∞) } , equipped with the metric