The Initial Value Problem for Singular SPDEs via Rough Paths

In this contribution we are interested in developing a solution theory for singular quasilinear stochastic partial differential equations subject to an initial condition. We obtain our solution theory via 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 observe that, thanks to bounds for the heat semigroup, enforcing an initial condition within the framework of Otto and Weber does not require any new stochastic results.