A Discrete Approach to Rough Parabolic Equations

Abstract

By combining the formalism of [8] with a discrete approach close to the considerations of [6], we interpret and we solve the rough partial differential equation $$dy_t=Ay_tdt+\sum_{i=1}^mf_i(y_t)dx_t^i, t\in[0,T]$$ on a compact domain $\mathcal{O}$ of $\mathbb{R}^n$, where  $A$ is a rather general elliptic operator of $L^p(\mathcal{O})$, $p>1$, and $f_i(\varphi)(\xi)=f_i(\varphi(\xi))$, 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.