A discrete approach to Rough Parabolic Equations

Abstract

By combining the formalism of \cite{RHE} with a discrete approach close to the considerations of \cite{Davie}, we interpret and solve the rough partial differential equation $d{y}_t=A{y}_{t}dt+{\sum }_{i=1}^m{f}_i(y_t)d{x}_t^i$ ($t\in [0,T]$) on a compact domain $\mathcal{O}$ of ${\mathbb{R}}^n$, where $A$ is a rather general elliptic operator of $L^p(\mathcal{O})$ ($p>1$), $f_i(\vp)(ξ):=f_i(\vp(ξ))$ and $x$ is the generator of a $2$-rough path. The (global) existence, uniqueness and continuity of a solution is established under classical regularity assumptions for $f_i$. Some identification procedures are also provided in order to justify our interpretation of the problem.