Lipschitz-stability of Controlled Rough Paths and Rough Differential Equations

Abstract

We provide an account for the existence and uniqueness of solutions to rough differential equations in infinite dimensions under the framework of controlled rough paths. The case when the driving path is $\alpha$-Hölder continuous for $\alpha>1/3$ is widely available in the literature. In its extension to the case when $\alpha\leqslant1/3,$ the main challenge and missing ingredient is to show that controlled rough paths are closed under composition with Lipschitz transformations. Establishing such a property precisely, which has a strong algebraic nature, is a main purpose of the present article.