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

TL;DR

This paper provides a unified proof of the Yamada-Watanabe-Engelbert theorem for various solutions of SPDEs in Banach spaces, extending classical results to new settings with flexible solution notions.

Contribution

It generalizes the Yamada-Watanabe theorem for SPDEs in Banach spaces, covering multiple solution types and function space settings, and constructs a measurable stochastic integral representation.

Findings

Extends Yamada-Watanabe theorems to Banach spaces with martingale type 2 or UMD.

Provides a measurable representation of stochastic integrals in Banach spaces.

Includes solutions in various frameworks like strong, weak, and mild solutions.

Abstract

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 $\omega$.