The Initial Value Problem for Singular SPDEs via Rough Paths

Abstract: In this contribution we develop a solution theory for singular quasilinear stochastic partial differential equations subject to an initial condition. We obtain our solution theory as a perturbation of the rough path approach developed to handle the space-time periodic problem by Otto and Weber (2019). As in their work, we assume that the forcing is of class $C^{\alpha -2}$ for $\alpha \in (\frac{2}{3},1)$ and space-time periodic and, additionally, that the initial condition is of class $C^{\alpha}$ and periodic. We contribute to the analytic aspects of the theory. Indeed, we show that we can enforce the initial condition via correcting the previously obtained space-time periodic solution with an initial boundary layer which may be handled in a completely deterministic manner. Uniqueness is obtained in the class of solutions which are corrected in this way by an initial boundary layer. Moreover, stability of the solutions with respect to perturbations of the data is established.