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

TL;DR

This paper establishes the well-posedness and stability of semi-linear rough partial differential equations on , introducing new energy estimate techniques in weighted spaces to handle degeneracy and nonlinearity.

Contribution

It develops a variational approach for rough PDEs, proving well-posedness and stability, and introduces a novel energy estimate method for non-degenerate linear rough PDEs.

Findings

Proves well-posedness of linear and semi-linear rough PDEs in .

Introduces a new energy estimate method using weighted L^2 spaces.

Extends well-posedness results to semi-linear perturbations.

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.