Solutions and approximations of some Lévy-driven stochastic (partial) differential equations

In this work we look at solutions to stochastic partial differential equations (SPDEs) with noise induced by a Lévy process in the context of Marcus integrals. The canonical Marcus integral is known from the study of SDEs with Lévy noise. We recapture the fundamental results on the existence of solution flows to the Marcus SDE and the convergence of Wong-Zakai approximations. We also prove a generalized Itô formula for said solutions and use this result to establish equations for the inverse flow. We are then looking at extensions of Marcus integrals to the case of SPDEs and find solutions for these equations. Our focus mainly lies on multi-dimensional first-order transport equations driven by Lévy noise. Existence and uniqueness results for the Marcus SPDE are established using a method of characteristics. For second-order equations we prove the existence and uniqueness of mild solutions for equations driven by pure jump Lévy processes, also in terms of Marcus SPDEs. Finally, we study a one-dimensional second-order advection-diffusion equation on the half-line, with Lévy noise at the boundary. Both Dirichlet and Neumann boundary conditions are considered, and the closed form formulae for mild solutions are determined. We also define Wong-Zakai type approximations of the solution by classical solutions and show convergence in the setting of the M1-topology in the Skorokhod space.