An energy method for rough partial differential equations

TL;DR

This paper develops an energy-based framework for establishing well-posedness and stability of nondegenerate linear parabolic PDEs driven by rough paths, extending classical PDE theory to rough stochastic settings.

Contribution

It introduces a notion of weak solution satisfying energy estimates directly derived from the rough PDE, and proves existence and uniqueness under optimal assumptions.

Findings

Weak solutions satisfy intrinsic energy estimates.

Existence shown via compactness of approximate solutions.

Uniqueness established through doubling variables and diagonal passage.

Abstract

We present a well-posedness and stability result for a class of nondegenerate linear parabolic equations driven by rough paths. More precisely, we introduce a notion of weak solution that satisfies an intrinsic formulation of the equation in a suitable Sobolev space of negative order. Weak solutions are then shown to satisfy the corresponding energy estimates which are deduced directly from the equation. Existence is obtained by showing compactness of a suitable sequence of approximate solutions whereas uniqueness relies on a doubling of variables argument and a careful analysis of the passage to the diagonal. Our result is optimal in the sense that the assumptions on the deterministic part of the equation as well as the initial condition are the same as in the classical PDEs theory.