The Yamada-Watanabe Theorem for mild solutions to stochastic partial differential equations
Stefan Tappe
TL;DR
The paper extends the Yamada-Watanabe theorem to mild solutions of semilinear SPDEs with path-dependent coefficients by employing the method of moving frame to reduce the SPDE to a Hilbert-space valued SDE. It then leverages the infinite-dimensional Yamada-Watanabe theorem for SDEs and a careful back-and-forth translation between the SPDE and the SDE to establish an equivalence: a unique mild solution exists iff martingale solutions exist for all initial laws and pathwise uniqueness holds. The contributions include a rigorous reduction framework, a complete proof strategy, and an illustrative example showing applicability to Hölder-type nonlinearities with compact semigroups. This work provides a robust tool for proving strong well-posedness of a broad class of SPDEs with path dependence, with potential impact on stochastic analysis and applications requiring reliable solution properties in infinite dimensions.
Abstract
We prove the Yamada-Watanabe Theorem for semilinear stochastic partial differential equations with path-dependent coefficients. The so-called "method of the moving frame" allows us to reduce the proof to the Yamada-Watanabe Theorem for stochastic differential equations in infinite dimensions.