Some refinements of existence results for SPDEs driven by Wiener processes and Poisson random measures
Stefan Tappe
TL;DR
The paper studies semilinear SPDEs on Hilbert spaces driven by Wiener processes and Poisson random measures, establishing existence and uniqueness of local and global mild solutions under local Lipschitz and linear-growth (or local-boundedness) conditions. By the moving frame method, SPDEs are reduced to Hilbert-space SDEs, enabling a rigorous link between strong SDEs and mild SPDEs and ensuring càdlàg trajectories. It proves local and global results for SDEs under local Lipschitz and growth assumptions, and extends these results to SPDEs via a dilation framework that does not require analyticity of the semigroup. The work contrasts with Cao’s successive-approximation approach, showing complementary regimes of applicability and providing a robust framework for SPDEs with jumps on general pseudo-contractive semigroups.
Abstract
We provide existence and uniqueness of global (and local) mild solutions for a general class of semilinear stochastic partial differential equations driven by Wiener processes and Poisson random measures under local Lipschitz and linear growth (or local boundedness, resp.) conditions. The so-called "method of the moving frame" allows us to reduce the SPDE problems to SDE problems.