The non-linear sewing lemma I : weak formulation

TL;DR

This paper introduces a novel framework for rough differential equations using flows, establishing existence of measurable flows under weak conditions and conditions for unique Lipschitz flows, connecting to sewing lemmas and Euler schemes.

Contribution

It presents a new approach to rough differential equations based on flows, with proofs of existence, uniqueness conditions, and links to existing sewing lemmas and numerical schemes.

Findings

Existence of measurable flows under weak conditions.

Conditions for unique Lipschitz flows.

Connection to sewing lemmas and rough Euler scheme.

Abstract

We introduce a new framework to deal with rough differential equations based on flows and their approximations. Our main result is to prove that measurable flows exist under weak conditions, even solutions to the corresponding rough differential equations are not unique. We show that under additional conditions of the approximation, there exists a unique Lipschitz flow. Then, a perturbation formula is given. Finally, we link our approach to the additive, multiplicative sewing lemmas and the rough Euler scheme.