A discrete approach to Rough Parabolic Equations

TL;DR

This paper develops a discrete method to interpret and solve rough parabolic equations involving elliptic operators and rough paths, establishing existence, uniqueness, and continuity of solutions under classical regularity conditions.

Contribution

It introduces a novel discrete approach to rough partial differential equations, extending formalism and providing rigorous existence and uniqueness results.

Findings

Established global existence and uniqueness of solutions.

Provided a framework for interpreting rough PDEs with elliptic operators.

Ensured solution continuity under classical regularity assumptions.

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 $dy_t=A y_t \, dt+\sum_{i=1}^m f_i(y_t) \, dx^i_t$ ($t\in [0,T]$) on a compact domain $\mathcal{O}$ of $\R^n$, where $A$ is a rather general elliptic operator of $L^p(\mathcal{O})$ ($p>1$), $f_i(\vp)(\xi):=f_i(\vp(\xi))$ 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.