Existence, uniqueness and stability of semi-linear rough partial differential equations

Abstract

We prove well-posedness and rough path stability of a class of linear and semi-linear rough PDE's on ${\mathbb{R}}^d$ using the variational approach. This includes well-posedness of (possibly degenerate) linear rough PDE's in $L^p({\mathbb{R}}^d)$, and then -- based on a new method -- energy estimates for non-degenerate linear rough PDE's. We accomplish this by controlling the energy in a properly chosen weighted $L^2$-space, where the weight is given as a solution of an associated backward equation. These estimates then allow us to extend well-posedness for linear rough PDE's to semi-linear perturbations.