An It\^o Formula for rough partial differential equations and some applications

We investigate existence, uniqueness and regularity for solutions of rough parabolic equations of the form $\partial _tu-A_tu-f=(\dot X_t(x) \cdot \nabla + \dot Y_t(x))u$ on $[0,T]\times\mathbb{R}^d.$ To do so, we introduce a concept of "differential rough driver", which comes with a counterpart of the usual controlled paths spaces in rough paths theory, built on the Sobolev spaces $W^{k,p}.$ We also define a natural notion of geometricity in this context, and show how it relates to a product formula for controlled paths. In the case of transport noise (i.e.\ when $Y=0$), we use this framework to prove an It\^o Formula (in the sense of a chain rule) for Nemytskii operations of the form $u\mapsto F(u),$ where $F$ is $C^2$ and vanishes at the origin. Our method is based on energy estimates, and a generalization of the Moser Iteration argument to prove boundedness of a dense class of solutions of parabolic problems as above. In particular, we avoid the use of flow transformations and work directly at the level of the original equation. We also show the corresponding chain rule for $F(u)=|u|^p$ with $p\geq 2,$ but also when $Y\neq 0$ and $p\geq 4.$ As an application of these results, we prove existence and uniqueness of a suitable class of $L^p$-solutions of parabolic equations with multiplicative noise. Another related development is the homogeneous Dirichlet boundary problem on a smooth domain, for which a weak maximum principle is shown under appropriate assumptions on the coefficients.