The Initial Value Problem for Singular SPDEs via Rough Paths

TL;DR

This paper develops a solution theory for singular quasilinear SPDEs with initial conditions, extending rough path methods to handle initial boundary layers and proving uniqueness and stability of solutions.

Contribution

It introduces a novel approach to incorporate initial conditions into the rough path framework for singular SPDEs, including deterministic handling of initial boundary layers.

Findings

Established a method to enforce initial conditions via boundary layer correction.

Proved uniqueness of solutions within the corrected class.

Demonstrated stability of solutions under data perturbations.

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.