The Yamada-Watanabe-Engelbert theorem for SPDEs in Banach spaces

We give a unified proof of the Yamada-Watanabe-Engelbert theorem for various notions of solutions for SPDEs in Banach spaces with cylindrical Wiener noise. We use Kurtz' generalization of the theorems of Yamada, Watanabe and Engelbert. In addition, we deduce the classical Yamada-Watanabe theorem for SPDEs, with a slightly different notion of `unique strong solution' than that corresponding to the result of Kurtz. Our setting includes analytically strong solutions, analytically weak solutions and mild solutions. Moreover, our approach offers flexibility with regard to the function spaces and integrability conditions that are chosen in the solution notion (and affect the meaning of existence and uniqueness). All results hold in Banach spaces which are either martingale type 2 or UMD. For analytically weak solutions, the results hold in arbitrary Banach spaces. In particular, our results extend the Yamada-Watanabe theorems of Ondrej\'at for mild solutions in 2-smooth Banach spaces, of R\"ockner et al. for the variational framework and of Kunze for analytically weak solutions, and cover many new settings. As a tool, and of interest itself, we construct a measurable representation $I$ of the stochastic integral in a martingale type 2 or UMD Banach space, in the sense that for any stochastically integrable process $f$ and cylindrical Brownian motion $W$, we have $I(f(\omega),W(\omega),\mathrm{Law}(f,W)) = (\int_0^{\cdot} f\, \mathrm{d}W)(\omega)$ for almost every $\omega$.