The non-linear sewing lemma II: Lipschitz continuous formulation

TL;DR

This paper develops a unified, Lipschitz continuous framework for solving rough differential equations using flows, extending previous approaches and establishing conditions for solution uniqueness and stability.

Contribution

It introduces tractable conditions for Lipschitz continuity of the flow and links the construction of solutions to perturbation formulas on almost flows.

Findings

Established conditions for Lipschitz continuity of the solution flow.

Linked the flow construction to solution uniqueness.

Provided perturbation formulas connecting different flow approximations.

Abstract

We give an unified framework to solve rough differential equations. Based on flows, our approach unifies the former ones developed by Davie, Friz-Victoir and Bailleul. The main idea is to build a flow from the iterated product of an almost flow which can be viewed as a good approximation of the solution at small time. In this second article, we give some tractable conditions under which the limit flow is Lipschitz continuous and its links with uniqueness of solutions of rough differential equations. We also give perturbation formulas on almost flows which link the former constructions.