Yamada-Watanabe uniqueness results for SPDEs driven by Wiener and pure jump processes
Kistosil Fahim, Erika Hausenblas, Kenneth H. Karlsen
TL;DR
The paper addresses well-posedness for SPDEs in variational form driven by both Wiener and jump (Poisson/Lévy) noise. It adapts Kurtz’s abstract framework to recast the SPDE in a Gelfand triple setting and proves that the existence of a martingale (weak) solution together with pathwise uniqueness implies the existence of a unique strong solution, thereby achieving uniqueness in law as a corollary. The main contributions are (i) extending Yamada-Watanabe uniqueness to infinite-dimensional Lévy-driven SPDEs, (ii) embedding the Itô-type stochastic integral with respect to Wiener and compensated Poisson noise into Kurtz’s framework, and (iii) proving a transequal lemma to transfer laws between weak and strong formulations. This work broadens the scope of robust well-posedness results for SPDEs with jump noise and provides a systematic route to obtain strong solutions from weak ones in variational settings with Lévy noise.
Abstract
The Yamada-Watanabe theory provides a robust framework for understanding stochastic equations driven by Wiener processes. Despite its comprehensive treatment in the literature, the applicability of the theory to SPDEs driven by Poisson random measures or, more generally, Lévy processes remains significantly less explored, with only a handful of results addressing this context. In this work, we leverage a result by Kurtz to demonstrate that the existence of a martingale solution combined with pathwise uniqueness implies the existence of a unique strong solution for SPDEs driven by both a Wiener process and a Poisson random measure. Our discussion is set within the variational framework, where the SPDE under consideration may be nonlinear. This work is influenced by earlier research conducted by the second author alongside de Bouard and Ondreját.